Category: Mathematics

Introduction

Set theory is a branch of mathematics that is based on logic, and it refers to the study of sets, i.e., groups of objects. The study of sets is most commonly used in the statistical field. One concept that can be used to better understand is the complement of sets, which refers to a group of objects that are not present in the given set. Studying complements of sets becomes much easier if we make use of De Morgan’s law. 

In this article, we will attempt to understand what complements of sets are, what De Morgan’s theorem is, and how it can be used to study complement of sets. It must be noted that De Morgan’s theorem is also a theorem in electronic circuits or Boolean algebra and the two must not be confused.

De Morgan’s Law

In set theory, De Morgan’s law refers to two transformation rules that govern complements of sets. It allows us to easily work with set theory since sometimes, calculations become much simpler and easier after applying De Morgan’s theorem.

De Morgan’s law of Union – (A∪B)’= A’ ∩ B’
This law states that the complement of the union of two sets is equal to the intersection of complements of the two sets. Graphically, this is represented in the figure below:

De Morgan’s law of intersection – (A∩B)’ = A’ ∪ B’
This law is simply the inverse of the previous theorem. It states that the complement of intersection of two sets is equal to the union of the complements of the two sets. You can see it in the figure below:

(A∩B)’ = A’ ∪ B’

What is the complement of a Set?

The complement of a set is defined as the set that includes all elements that are not present in A. That is, it can be considered to the opposite of a given set. If a set of fruits contains all bananas in the world, its complement will contain all fruits that aren’t bananas.

Graphically, consider a set A, which is a subset of the universal set U as shown below

Here, the complement of A can be defined as all elements that are not in A. Naturally, these elements will still be a part of U. Thus, we have the following definitions for the complement A’:

A’ = {x: x ⊄ A}
A’ = U – A

Complement of a Set Properties

There are a few properties of complements of sets that one must remember. These are listed below:

1. The union of a set with its complement yields the universal set. That is, A ∪ A’ = U.
2. The intersection of a set with its complement yields the null set Ø. Mathematically, A ∩ A’ = Ø.
3. The law of double complementation states that the complement of the complement of a set gives us the original set itself. (A’)’ = A.
4. The universal set contains all elements in the universe and the null set contains no elements. Thus, the universal and null sets are complements of each other. Therefore, Ø’ = U, and U’ = Ø.

Here are some examples to demonstrate these properties. Let U = {8, 9, 45, 89, 54, 25, 32} and A = {45, 54, 32}. Then, we see that A’ = {8, 9, 89, 25}. Therefore:

• A ∩ A’ = {45, 54, 32} ∩ {8, 9, 89, 25} = Ø (No common elements).
• (A’)’ = {8, 9, 89, 25}’ = {45, 54, 32}.

Solved Problems

1. Let U = {4, 38, 56, 100, 20, 12, 69}, A = {4, 56, 12}, and B = {20,69}. Prove De Morgan’s theorem for these sets.

Let us calculate the union of A and B. We get:

A ∪ B = {4, 56, 20, 12, 69}

(A ∪ B)’ = {4, 56, 20, 12, 69}’ = {38, 100}

Further,

A’ = {38, 100, 20, 69}

B’ = {4, 38, 56, 100, 12}

A’ ∩ B’ = {38, 100}

Similarly, for the second law, we have:

A ∩ B = Ø

(A ∩ B)’ = Ø’ = U

Now, 

A’ ∪ B’ = {4, 38, 56, 100, 20, 69, 12} = U

Thus, we have proved De Morgan’s law.

2. If U is the universal set containing students who passed in their exams, and A is a subset containing those students who passed in Physics. If U contains 100 elements, and A contains 30 elements, how many students failed in physics?

We know that A’, the complement of A, contains all students who failed physics. Thus, we have

A’ = U – A

A’ = 100 – 30 = 70

Therefore, 70 students failed physics.

3. If A is the set of odd numbers less than 20, U is the set of all natural numbers less than 21, what is the complement of A?

We can write

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}.

A = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}

The complement of A is all elements not in A but not in U. That is, A’ = U ∩ A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}.

 

Summary

This article was focussed on discussing the complements of sets, their properties, and De Morgan’s law. De Morgan’s law includes a pair of statements, which may be summarised by the rule of thumb “break the line, change the sign”. These rules are particularly useful for studying set complements.

Frequently Asked Questions

1. What will be the intersection of two null sets?

The intersection of two null sets leads to the null set itself.

2. What will be the union of a set with itself?

Union is defined as the collection of all elements that are present in both the sets and thus, the union of a set with the set itself leads once again to the set itself.

3. What is the relation between the set of integers and the set of whole numbers?

Integers contain all negative numbers as well while whole numbers do not contain negative numbers. Thus, the set of whole numbers is a subset of the set of integers.

4. Do sets obey commutative, additive, or distributive laws?

Yes. These laws are defined in the concepts of set algebra and are followed by sets in general.

Introduction

Fractions generally do not have any restriction on the numbers that appear in their numerator or denominator, apart from the fact that the denominator can be zero. However, fractions whose denominators are powers of 10 belong to a specific class of fractions known as decimal fractions. These fractions are most often encountered in scientific calculations, architectural designing, fitness parameters, etc. Hence, their uses are vast in number, and we must have a functional understanding of such fractions. This article is aimed at providing the same.

Decimal Fractions

The word “decimal fractions” is made up of two mathematical words, viz. “decimal” and “fractions”. Separately, we are aware that “decimal” means something that is connected to the value 10 (deci means 10), and that “fraction” refers to ratios. Together, decimal fractions are those fractions wherein, the denominator is a power of 10. Some easy examples of decimal fractions are 21/10, 53/100, 34/1000. As we will soon see, decimal fractions have some special properties that allow us to classify them together.

Just like other fractions, decimal fractions can also be represented in the form p/q. The only difference is that this time, q is necessarily a power of 10. In the figure below, we graphically represent the difference between decimal and non-decimal fractions.

Place value of Decimal Fractions

When studying whole numbers, we have come across place values, which increase towards the left. The right-most digit has the place value of one, then ten, then hundred, and so on. Since decimal fractions contain powers of 10 in the denominator, their value decreases if the power of 10 decreases. 

Take a look at the figure below and notice how with increasing power of 10, the fraction becomes smaller and smaller. We start with tenths, then we get hundredths, then thousandths, and so on. By tenths, we mean one in ten portions and thus, one hundredth is much smaller than one tenth.

Operations on Decimal Fractions

All fractions can undergo addition, subtraction, multiplication, and division, and decimal fractions are no different. The rules to be followed remain the same here. However, since the denominator is always a power of 10 only, calculations are simpler this time around.

Conversion to Decimal Fractions

The conversion of any number to a decimal fraction is an easy task and we will discuss it here. 

First, we have the simplest case of a number carrying a decimal point. Given such a number, we simply need to remove the decimal point and put as many powers of 10 in the denominator as there were digits after the decimal points. For example, 3.14 is simply 314/10² = 314/100.

Next, if we are given a fraction, we check the denominator and see if it can be directly converted to a power of 10 by multiplication with a suitable factor. For instance, given the number 13/20, we see that we can multiply 20 with 5 to get 100. Thus, we multiply the numerator and denominator with 5 to get 65/100, which is the decimal fraction form of 13/20.

If that is also not possible, we simply divide the fraction to get a decimal number and convert it into a decimal fraction as previously stated. For example, 10/21 can’t be directly converted to a decimal fraction. Thus,

10/21 = 0.48 (approximately)

And 0.48 = 48/100.

Addition of Decimal Fractions

The addition of decimal fractions can be achieved via two ways:

1. Convert to a decimal number and add the decimal numbers.
   For example, given 7/10 and 89/100, we get the decimal forms as 0.7 and 0.89. Now 0.7+0.89 = 1.59 = 159/100.
2. Make the denominators of the two numbers same and then add.
   Taking the same example as before, 7/10 and 89/100 can be given the same denominator if we multiply and divide the first fraction by 10. Thus, we have 7/10 = 70/100. Now, we can easily add these numbers to get 70/100 + 89/100 = 159/100. As expected, the answer is the same.

Subtraction of Decimal Fractions

The subtraction of decimal fractions follows exactly the same procedure as addition. Thus, 62/100 – 1/10 = 0.62-0.1 = 0.52 = 52/100. The same result may be achieved by making the denominators the same.

Multiplication of Decimal Fractions

Multiplication for decimal fractions follows the same rules as normal fractions. Thus, the numerator is multiplied with the numerator and the denominator with the denominator. Since the denominators are both powers of 10, we just add their powers to get the product for the denominator. Thus, 9/10 ✕ 178/1000 = (9 ✕ 178) / (10 ✕ 1000) = 1602/10000.

Division of Decimal Fractions

Just like other operations, division of decimal fractions also follows the same procedure. However, since the denominators of the two fractions are both powers of 10, we can easily subtract the powers for the denominators. In general, to divide two fractions, we first write the reciprocal/inverse of the second fraction and multiply it with the first one.

For example, given 49/100 and 134/1000, we have 

Types of Decimal Fractions

Decimal fractions can be categorised into three types, which are listed below:

• Terminating decimals: If the decimal number form of the fraction ends after a finite number of digits, the fraction is said to be terminating. For example, 49/100 = 0.49, 45/10 = 4.5, etc.
• Non-terminating repeating decimals: These are decimal fractions which, when represented in decimal form, repeat their digits indefinitely. For example, 2/11 = 0.1818181818…
• Non-terminating non-repeating decimals: These are irrational numbers whose decimal expansion does not end and does not repeat. The best example is the approximation for π = 22/7 = 3.1415926535897…

Solved Examples on Decimal Fractions

1. Convert 5/16 to a decimal fraction

We see that the denominator is equal to 16. If we multiply 16 with 625, we get 16 ✕ 625 = 10000. Thus, 

5/16 = (5✕625) / (16✕625) = 3125/10000.

2. If a tank contains 4.89 litres of oil and 2.62 litres are removed from it, how many litres are left. Use decimal fractions.

Our task is to first convert these numbers to decimal fractions and then subtract. Hence,

4.89 = 489/100

2.62 = 262/100

489 / 100 – 262 / 100 = (489-262) / 100 = 227/100 = 2.27

Hence, the tank now contains 2.27 litres of oil.

3. The product of two numbers is 111.09 and one of them is 6.9. Find the other number.

Let us again convert the numbers to decimal fractions to get

111.09 = 11109/100

6.9 = 69/10 = 690/100

To find the unknown number, we will need to divide 11109/100 with 690/100. Hence, if the unknown number is x, we have

x = (11109/100) / (690/100)

X = 11109/100 ✕ 100/690 = 11109/690 = 16.1

Thus, the other number is 16.1

Summary

This article briefly summarised the concepts of decimal fractions and algebraic operations related to these fractions. We learnt that decimal fractions are a special class of fractions whose denominators are powers of 10 only. We further learnt how to represent any number in the form of a decimal fraction. Hence, this article serves as a useful guide for understanding the concept of decimal fractions.

Frequently Asked Questions

1. What is the difference between decimal fractions and percentages?

Decimal fractions can contain any power of 10 in their denominator. However, percentages are always calculated out of 100 and thus, the denominator is necessarily 100.

2. How do we round off decimal numbers?

Rounding off decimal numbers is similar to rounding off whole numbers. We start with the last digits and check whether it is greater than equal to 5. If so, we increase the second-last digit by 1 and drop the last digit. If not, we simply drop the last digit without incrementing the second-last one. This process is continued till the desired number of digits are reached.

3. Are all rational numbers decimal fractions?

All rational numbers can be represented in the form of decimal fractions. However, this does not mean they are decimal fractions. But the converse is true. All decimal fractions are necessarily rational numbers.

4. Do decimal fractions obey commutative, additive, or distributive laws?

Yes. Decimal fractions obey all the laws that govern normal fractions.

Introduction

A dataset is a collection of data that is organised and stored in a set. To ease understanding and description, the data is generally arranged in the form of tables where each column represents a distinct variable. You must have come across datasets numerous times in your daily routines. For instance, the attendance register of a class is an example of a dataset. Depending on the situation, each row in the table corresponds to an entry in the dataset for which the values of different attributes are listed under their corresponding columns. A complete dataset contains the values of each attribute for each of its members.

In this article, we will delve into the definition of a dataset, the different types of datasets, their properties, and provide solved examples to aid in understanding.

What is a Dataset?

As mentioned, a dataset is a collection of data or observations from experiments, measurements, calculations, etc., that is organised in a specific manner. This data can be of any kind like names, numbers, figures, description, etc. Further, it is not necessary to present the data in tabular format and one can also present it via charts and graphs.

Generally, a single dataset groups together related values and objects. The easiest example would be that of the list of students in a class and their attendance record. This dataset would be organised in the form of rows and columns, where each row could correspond to a student and each column would represent his attendance status on a given date.

Types of Datasets

Since the type of data that we wish to present can vary in nature, datasets are classified into various types as follows:

• Numerical Dataset
• Bivariate Dataset
• Multivariate Dataset
• Categorical Dataset
• Correlation Dataset

We will discuss each of these types with examples.

Numerical Dataset

This is the simplest form of dataset containing numerical data. There are no words or pictures in a numerical dataset and one can also say that numerical datasets contain quantitative data. Since all entries in a numerical dataset are numbers, we can easily apply various arithmetic operations on any entry in the dataset. Some common example of such a dataset would be: 

• Age of set of people
• Number of balls played
• Number of shoes in the shoe rack

Bivariate Dataset

The word bivariate is a combination of bi, meaning two, and variate, referring to variables. That is, a bivariate dataset contains two variables which generally have some sort of relationship with each other. The value of the second variable depends on the value of the first one.
If you have ever seen a table listing the number of calories you would burn against the time you work out, that is precisely what a bivariate dataset is. Naturally, the number of calories burnt increases with the workout time.

Multivariate Dataset

A multivariate dataset is akin to a bivariate one, except that it contains more than two variables. Generally, the variables in such a dataset are functions of one or more variables and thus, each column is related to some other one.

For example, a dataset that lists the price of different dishes across different restaurants of your city is an excellent example of a multivariate dataset. Not only does the cost of dishes vary with restaurants, but if the dish in question (say pizza) contains topping, its cost would depend on the cost of that topping in that particular restaurant, leading to a complex, intricately linked dataset.

Categorical Dataset

Generally, categorical datasets contain qualitative data like a person or object’s attributes or characteristics. In datasets, if a variable can take any one of two values, it is said to be dichotomous. On the other hand, if a variable can take one of a large number of values, it is said to be polytomous. Some examples of a categorical dataset would be datasets storing a person’s hair length (short or long) or the type of different cars (automatic or manual).

Correlation Dataset

Correlation datasets contain data whose variables have a relationship with each other and thus, are interdependent. These relationships may be of the following nature:

• Positive correlation: The variation in the variables occurs in the same direction. That is, if one of the variables increases, the other one also increases and vice versa. For instance, a jogger’s distance covered can only increase with time, not decrease.
• Negative correlation: Negatively correlated variables vary in opposite directions. If one of them increases, the other one decreases. For example, the time taken to cover a distance of 10 miles is sure to decrease if the speed of the car increases.
• No correlation: It is also possible for variables to be totally independent of each other. For instance, the number of flowers in a park generally has nothing to do with the number of flowers in it.

Mean, Median, Mode, and Range

Mean, median, mode, and range are quantities used to investigate the nature, variation, and properties of a dataset. Mean, median, and mode are often referred to as measures of central tendency, which means that they describe where the centre of a collection of values lies. We will discuss these topics here.

• Mean: Mean refers to the average value of a variable. Mathematically, mean is calculated by dividing the sum of all the values of a variable and dividing it by the number of observations. For example, if the students in a class scored {40, 50, 60, 40, 60} marks out of 100, then the average score of the class would be calculated as follows:
  Mean = sum of observations / number of observations
  Mean = (40+50+60+40+60) / 5 = 250/5 = 50
• Median: Median refers to the central value in a collection of values that has been sorted in ascending or descending order. It is important for the data to be sorted or the median calculated would be incorrect. For example, if we are given the values {1, 6, 5, 7, 2}, then in ascending order, we have {1, 2, 5, 6, 7}. Here, 5 is the value that lies in the centre of the datasets and thus, is the median.
• Mode: The entry that occurs the most frequently in a collection of values is known as the mode of that collection. For example, out of {1, 2, 6, 5, 3, 3, 4, 5, 2, 3, 7, 8, 6, 3, 6, 9, 3}, 3 occurs the most frequently and thus, is the mode.
• Range: The range is indicative of the spread of a variable. It is mathematically calculated by subtracting the smallest value from the largest one in a collection. Thus, in {1, 2, 4, 5, 6}, the range is given by R = 6-1 = 5.

Properties

There are a number of properties related to data analysis that can help us understand the dataset in question. Depending on these properties, we can choose the best method of statistical analysis to be applied. These properties are analysed via a process known as exploratory data analysis (EDA) and some of them are listed here:

• The centre of the data.
• The skewness of data.
• Distance between data members.
• Presence or absence of outliers.
• Correlation between variables.
• The probability distribution type of the variable. 

Examples

Determine the mean, median, mode, and range of {1, 11, 7, 3, 9, 3, 15}.

It is generally a good practice to sort the data in ascending or descending order from the offset. Thus, we have {1, 3, 3, 7, 9, 11, 15}. Now, we have 

Mean = sum of all observations / number of observations.
Mean = (1+3+3+7+9+11+15) / 7 = 49/7 = 7.

Median = 4th value = 7.

Mode = Most frequent value = 3

Range = highest value – lowest value = 14.

Summary

A dataset is a collection of data organized in a specific manner and it is classified into various types like numerical, bivariate, multivariate, categorical, and correlation types. Datasets are studied via various values like mean, median, mode, and range.

Mean refers to the average value of a data set. Median is the central value after the dataset has been arranged in ascending or descending order. Mode is the most frequently occurring value while range describes the spread of data measured as the difference of highest and lowest value.

Frequently Asked Questions

1. What do you mean by a dataset?

A dataset is simply a collection of data organised in a specific manner like in a table.

2. Is the range also a measure of central tendency?

No. Mean, median, and mode are the true measures of central tendency. Range is a useful value, but it isn’t classified as a central tendency measure.

3. How would you find the mode of a dataset which contains no distinct values, i.e., all values are the same?

For such a dataset, the mode would be this repeating value itself. Note that if no value in the dataset repeats, then the mode would be undefined.

An introduction

Graphs of many kinds may have appeared in publications and newspapers. Data is analyzed for a variety of purposes. We can identify kids who are overweight or underweight, for instance, if we gather information on the weights of the various students in a class. With the use of this information, we can suggest healthy eating to pupils who are underweight. Based on these facts, we can also create a chart. To understand data, many graphs are utilized. There are many uses for a bar graph, which is a sort of graph used to depict data. They are two-dimensional illustrations. Bars of varied heights are a useful tool for representing data. They typically serve to represent statistical and economic data. Let’s look at the chapter’s bar graph.

Definition of Bar Graph

A bar graph is a graph in which data is displayed as bars, with the height of the bars depending on the values they stand for. The bar may be vertical or horizontal. Bar charts are another name for them. Bar charts are used to compare independent variables. They are quite helpful in comprehending facts. A bar chart does not need that the data be in any particular arrangement. An x-axis and a y-axis with titles and labels are both present in a bar chart.

Bar Graph – Properties

• The width of each bar should be the same.
• Both horizontal and vertical bars are possible.
• The data the bar represents determines its height.
• A bar chart’s bars should all be based on the same line.

Types of Bar Graphs

There are two main categories of bar graphs:

• Vertical Bar Graph
• Horizontal Bar Graph

In addition to the above two categories, bar graphs can be categorized as follows:

• Grouped Bar Graph
• Stacked Bar Graph

Let us see these types of Bar Graphs in detail.

Vertical Bar Graph

Rectangular bars in a vertical bar graph are located along the y-axis and run vertically. All of the bars’ bases are along the x-axis. The values the bars stand for determine the heights of the bars.

Horizontal Bar Graph

All rectangular bars in this style of the bar graph are located on the x-axis. All of the bars’ bases are along the y-axis. The values the bars stand for determine the lengths of the bars.

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Stacked Bar Graph

In a stacked bar graph, different parts are represented by different colored bars. The composite bar chart is the name of it. Different parts of a bar stand for various categories. They are utilized to add more representational parameters.

Grouped Bar Graph

The clustered bar graph is another name for this form of a bar graph. Data can initially be categorized according to a main variable. Using a secondary variable, we may compare the information in each set. This kind of graph can employ both vertical bar charts and horizontal bar charts.

Types of Bar Graphs

Differences between Bar Graph and Histogram

Steps to draw a Bar Graph

Let’s use a class of 50 students as an illustration. There are15,6,10, and 19 students who arrive by bicycle, auto, foot, and bus, respectively.

1. Determine the chart’s variables.
2. Select the type of chart.
3. Give the graph a name.
4. Choose a date range for each axis.
5. Sketch the x- and y-axes.
6. Indicate the number of pupils on the x-axis.
7. Indicate the mode of transportation on the y-axis.
8. Draw the bar graph for each category based on the values.

Interesting facts

• Bar graphs offer a straightforward way to visualize complicated data.
• We can understand the data trend easily.
• Bar graphs are frequently used in both scientific and business reports.

Bar Graphs Solved examples:

1. A business has locations in five cities. The information about the branches’ sales for August is shown below. Using these details as a foundation, create a bar graph.

2. In a classroom, there was a sketching contest. Following are details about the amount of pupils and the colours they chose to use. Based on these facts, create a bar graph.

Conclusion

The definition of a bar graph, the distinctions between a bar graph and a histogram, and examples of several types of bar graphs were covered in this chapter.

                                                                                                                           

Frequently Asked Questions 

1. Explain the differences between a component bar chart and a percentage bar chart.

 Each rectangle bar is broken down into different sub-components in a component bar chart. In a percentage bar chart, all bars have a magnitude of 100, and different components are displayed according to their percentage.           

2. What are a stacked bar graph’s benefits and drawbacks?

Each rectangular bar in a stacked bar chart represents a key category. Subcategories are used to categorise each bar. It is used to compare different study parameters. This type of chart’s drawback is that it is challenging to grasp.

3. When would a bar graph be preferable to a line graph?

Line diagrams are helpful for showing more subtle alterations to a pattern over time. For analysing more significant shifts or differences in data between groups, bar graphs are preferable.

Introduction

Data refers to information about a single or multiple entities, and organizing it effectively is crucial to understand the behaviour of a system. This is where data handling, a statistical analysis, comes into play. It involves collecting, regrouping, analysing, and presenting data through graphs, charts, and other visual aids. When we create graphs in presentations for our projects or say that 83% of voters liked ice cream over milk, we are engaging in data handling.

The main goal of data handling is to ensure the safe and orderly storage of data, making it easily accessible when needed. Voter polls, online surveys, and population statistics are real-life examples of data handling in action.

Data Handling Definition

Data handling is crucial for effectively managing research data and preventing unnecessary obstacles in accessing specific information. For example, if you have records of when you studied a particular topic and your notebooks are organised by date, you will be able to quickly find the notes for that particular topic.

Data handling involves organizing and securely storing data in electronic or non-electronic formats, such as desktops, CDs, notebooks, and journals. By properly handling data, organizations can present critical information accurately and predict the behaviour of the parameters involved in the data. 

What is Data Handling

Data refers to a single or a group of information regarding an event or multiple events. To facilitate further analysis and research, it is crucial to organize and document the data effectively. Different statistical methods can be used to represent the data accurately, including bar graphs, line graphs, pie charts, histograms, and more. 

For example, a data set representing the spending behaviour of different groups of people can be graphically represented through a bar graph that shows the spending amount (%) from their household savings, as seen in the figure below. Looking at the graph, you can easily infer which classes of people had the most spendings and which classes preferred to increase their savings. Data represented in visual format is far easier to understand than numbers and thus, data handling is an important statistical process.

Bar Graph

Collection of Data

The data collection process is a critical step in data handling. The data to be analysed must be collected accurately and in a way that allows for proper grouping and analysis. For example, if a teacher wants to know the attendance record of each student in a particular class, they will need to gather attendance data from the register for each subject taught in that class. By calculating the attendance percentage of each student in each subject, the teacher can determine the attendance status of each student. 

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However, imagine if the teacher accidentally included data from another class. An even more likely error can occur because it is easy to mix up the cells in a table and thus, wrong attendance can be marked. This will lead to incorrect calculations that will be of no use and hence, it is essential to collect data while ensuring correctness.

The Importance of Ensuring Accuracy and Appropriateness

While collecting data, integrity is an essential component, and a data set is of best use when it is error-free and accurate. Otherwise, difficulties can arise in analysing it and performing analysis. Various factors related to ensuring accuracy are summarised below:

The Consequences of Improperly Collected Data

Various problems can arise when data is improperly or incorrectly collected. Here are a few:

• The data doesn’t line up with real-world scenarios and results and thus, the organisation deteriorates.
• Collecting unwanted or random data increases storage cost but has no benefit.
• The person conducting research on the data may get misguided due to incorrect data and provide wrong results.
• Incorrect data means incorrect predictions, which would lead to us making the wrong decisions.
• Incorrect data is invalid and cannot be reused again, increasing costs.

Methods

There are a number of data collection methods, which are summarised below:

Organization of Data

Organizing data accurately is a crucial aspect of data handling. It requires a systematic and accessible approach so that future readers of the data can obtain the desired results easily. The Oxford English Dictionary is an excellent example of effective organisation of a large set of data. With hundreds of thousands of words in the English language organised and grouped by alphabet, it is easy to find any word we want since they are ordered by the alphabet. Similarly, telephone books, global maps, and various other real-life data sets rely on effective organization and representation of large sets of data.

Types of Data

Classifying data helps us decide the type that will work best for our use-case. Data can be broadly divided into two types: qualitative and quantitative data. 

Qualitative data provides descriptive information and characteristics about the variable in question. It can include how something is changing, if there are sudden dips or rises, etc. 

On the other hand, quantitative data represents numerical or algebraic values like the average rainfall, average temperature, etc. Quantitative data itself may be discrete or continuous in nature.

Data Graphs

How to Represent Data

Data can be represented in a large number of ways, chief among which are:

• Bar graph
• Line graph 
• Pictograph
• Histogram
• Stem and leaf plot
• Dot plots
• Frequency distribution
• Cumulative tables and graphs

Summary

In this article, we discussed an overview of data handling, including its advantages, types of data collection, and the consequences of incorrect data handling. We understood how and why data handling is a crucial component in various industries and management services. Finally, we summarised the various types and representations of data.

Frequently Asked Questions 

1. Which type of graph represents data with pictures?

A pictograph includes the pictures to represent the data.

2. Are charts and graphs the same?

No. While all graphs can be considered as charts, the reverse may not be true.

3. Name graphical presentations of data other than bar graphs.

Data can be graphically represented via pie charts, histograms, scatter plots, area plots, etc.

4. What is the category of interview method?

Collecting data using interviews can be done either personally, or telephonically.

5. Out of qualitative and quantitative data, which is more effective?

For a short time period, quantitative data is cheaper and thus, is effective. However, over a large time interval, qualitative data becomes more helpful.

Introduction

A quadrilateral that can be completely inscribed in a circle is called a cyclic or inscribed quadrilateral and conversely, a circle passing through all four vertices of a quadrilateral is known as a circumcircle. The centre of such a circle is called the circumcentre and the radius is known as the circumradius. Another way of saying that a quadrilateral is cyclic is to say that its vertices are concyclic. Interestingly, while you can inscribe all triangles into a circle, the same is not possible with all quadrilaterals and instead, only some of them can be cyclic.

Definition – What is cyclic quadrilateral

A quadrilateral with all its vertices lying on a circle is called a cyclic quadrilateral. However, not every quadrilateral can be inscribed in a circle and thus, the quadrilateral must be cyclic by design.

The figure below shows a cyclic quadrilateral EFGH inscribed inside a circle. Its sides are represented by e, f, g, and h, and the diagonals are represented by p and q. Note that the diagonals need not be of equal length.

A cyclic quadrilateral

Angles

Angles opposite to each other inside a cyclic quadrilateral sum up to 180⁰, i.e., they are supplementary. For instance, in the figure shown above, we have a cyclic quadrilateral EFGH. If the angles made at the vertices of this quadrilateral are represented by ∠E, ∠F, ∠G, and ∠H, respectively, then we can write the following:

\(\angle E + \angle G = {180^0}\)

\(\angle F + \angle H = {180^0}\)

Further, just like all other quadrilaterals, the sum of all the angles of a quadrilateral is equal to 360⁰ and this can be easily proven by adding the two equations written above.

Radius

There are a few other interesting properties related to the side lengths of a cyclic quadrilateral. Given the cyclic quadrilateral EFGH as above, we can write the following:

semi perimeter of circumcircle, s = \(\frac{{e + f + g + h}}{2}\)

Radius of circumcircle\(r = \frac{1}{4} \times \sqrt {\frac{{(eg + fh) \times (eg + fh) \times (eh + fg)}}{{(s – e) \times (s – f) \times (s – g) \times (s – h)}}} \)

Diagonals

Once again, we look at the cyclic quadrilateral we saw above, with diagonals represented by p and q. Another interesting property that emerges is between the side lengths and diagonals of such a quadrilateral. We can write the following:

length of diagonal p \( = \frac{{(eg + fh) + (eh + fg)}}{{(ef + gh)}}\)

length of diagonal q \( = \frac{{(eg + fh) + (ef + gh)}}{{(eh + fg)}}\)

Area

We can also examine properties related to the area of cyclic quadrilaterals. Looking at the figure shown before, if we have the semi perimeter given by s, we can write the following:

semi perimeter s \( = \frac{{e + f + g + h}}{2}\)

Area \( = \sqrt {(s – e) \times (s – f) \times (s – g) \times (s – h)} \)

Theorems

Ptolemy’s theorem: This is an interesting theorem related to cyclic quadrilaterals. Let us discuss and prove it. As before, we have a cyclic quadrilateral represented by EFGH. Using Ptolemy’s theorem, which states that in cyclic quadrilateral, the product of the diagonals equals the sum of the products of pairs of two opposite sides. That is,

\((EF \times GH) + (EH \times FG) = EG \times FH\)

Or,

\({\bf{eg}} + {\bf{fh}} = {\bf{pq}}\)

This can be proven as follows. We take a cyclic quadrilateral ABCD and suppose that K is the point where its diagonals intersect. This is shown in the figure below.

Ptolemy’s theorem

Since the angle subtended by a chord are the same at any point on the circle, we can write for chord AD, chord BC, and chord AB,

∠ABD =∠ACD

∠BCA =∠BDA

∠BAC =∠BDC

Ptolemy’s theorem

Next, we take a point E on the diagonal AC such that ∠EBC = ∠ABD. From the previous three equations, we already have ∠BCA =∠BDA and thus, we have two similar triangles, namely, triangle EBC and triangle ABD. Thus, we can write the following:

\(\begin{array}{l}\frac{{CB}}{{DB}} = \frac{{CE}}{{AD}}\\CB \times AD = CE \times DB\end{array}\)

Let us consider this equation 1 and add ∠KBE on both sides of the equation. We then get

\(\begin{array}{*{20}{c}}{\angle {\bf{EBC}}{\rm{ }} + \angle {\bf{KBE}}{\rm{ }} = \angle {\bf{ABD}}{\rm{ }} + \angle {\bf{KBE}}}\\{\angle {\bf{KBC}}{\rm{ }} = \angle {\bf{ABE}}}\end{array}\)

Ptolemy’s theorem

Similarly, we can find similar triangles BDC and ABE and do something similar, leading us to the following relations:

\(\begin{array}{l}\frac{{AB}}{{BD}} = \frac{{AE}}{{DC}}\\DC \times AB = AE \times BD\end{array}\)

We call this equation 2 and add it to equation 1 to get

\(\begin{array}{l}CB \times AD + DC \times AB = CE \times DB + AE \times BD\\CB \times AD + DC \times AB = (CE + AE) \times BD\\CB \times AD + DC \times AB = AC \times BD\end{array}\)

And thus, we have our proof.

Properties

Let’s discuss some properties regarding cyclic quadrilateral:

1. We have discussed that the opposite angles of a cyclic quadrilateral are supplementary. This is always true and thus, if the sum of opposite angles of a quadrilateral is 180⁰, then the quadrilateral is necessarily cyclic.
2. A rhombus can never be a cyclic quadrilateral since its opposite angles do not sum up to 180⁰.
3. Given a cyclic quadrilateral EFGH, with side lengths e, f, g, and h respectively, with diagonals p and q, let the diagonals intersect at a point I. We can write

\(EI \times IG = FI \times IH\)

1. Joining the midpoints of the sides of a quadrilateral gives us a parallelogram.
2. The perpendicular bisectors of the sides of a cyclic quadrilateral meet at the centre and are concurrent.

Problems and Solutions

1. In a cyclic quadrilateral EFGH, \({\bf{if}}{\rm{ }}\angle {\bf{E}}{\rm{ }} = {\rm{ }}{\bf{8}}{{\bf{5}}^{\bf{0}}},{\rm{ }}{\bf{find}}{\rm{ }}\angle {\bf{G}}\)

Since the opposite angles of a cyclic quadrilateral are supplementary, we can write

\(\begin{array}{*{20}{c}}{\angle {\bf{E}}{\rm{ }} + {\rm{ }}\angle {\bf{G}}{\rm{ }} = {\rm{ }}{\bf{18}}{{\bf{0}}^{\bf{0}}}}\\{\angle {\bf{G}}{\rm{ }} = {\rm{ }}{\bf{18}}{{\bf{0}}^{\bf{0}}}–{\rm{ }}{\bf{8}}{{\bf{5}}^{\bf{0}}} = {\rm{ }}{\bf{9}}{{\bf{5}}^{\bf{0}}}}\end{array}\)

2. Let the side lengths e, f, g, and h of a cyclic quadrilateral be 3, 6, 4, and 7m respectively. What is its area?

For a cyclic quadrilateral, the semi perimeter is given by

\(s = \frac{{e + f + g + h}}{2}\)

And the area is given by

\(A = \sqrt {(s – e) \times (s – f) \times (s – g) \times (s – h)} \)

On substituting the values, we get s = 10 m. And therefore, the area is

\(\begin{array}{l}A = \sqrt {(10 – 3)(10 – 6)(10 – 4)(10 – 7)} \\A = \sqrt {7 \times 4 \times 6 \times 3} \\A = \sqrt {504} {m^2}\end{array}\)

3. Let the side lengths of a cyclic quadrilateral be 2, 5, 3, and 6m. Find the product of the diagonals.

We can use Ptolemy’s theorem to solve this problem. We know that

\(\begin{array}{l}EF \times GH + EH \times FG = EG \times FH\\EG \times FH = 2 \times 3 + 5 \times 6 = 36\end{array}\)

Summary

This article discussed what cyclic quadrilaterals are by explaining their definition and listed a few properties related to the sides, angles, the circumcircle, the diagonals, and the area of a cyclic quadrilateral. Further, we looked at a few theorems related to such quadrilaterals, namely, Ptolemy’s theorem.

Enhance your understanding of Cyclic Quadrilaterals by enrolling in our Class 9 Maths Tuitions.

Frequently Asked Questions

1. Is every square a cyclic quadrilateral?

Yes. The sum of the opposite angles inside a square always add up to 180⁰ and therefore, all squares are cyclic in nature.

2. If we are given the lengths of sides of a cyclic quadrilateral, how do we find its diagonals?

Such problems can be solved using the properties of cyclic quadrilaterals. The diagonals p and q of a cyclic quadrilateral EFGH can be obtained via the formulae given below:

length of diagonal p \( = \sqrt {\frac{{(eg + fh) + (eh + fg)}}{{(ef + gh)}}} \)

length of diagonal q \( = \sqrt {\frac{{(eg + fh) + (ef + gh)}}{{(eh + fg)}}} \)

3. Are all parallelograms cyclic quadrilaterals?

Not necessarily. The opposite angles inside a parallelogram aren’t always supplementary and thus, may not add up to 180⁰ , which means that only some parallelograms can be cyclic.

4. How can we prove that the opposite angles of a cyclic quadrilateral are supplementary?

A: We can prove this using the fact that the opposite angles of an inscribed angle are equal. Let ABCD be a cyclic quadrilateral with center O, and let angle ABD be x and angle BCD be y. Then, angle ABC is (180 – x) degrees and angle ADC is (180 – y) degrees, since angles on a straight line add up to 180 degrees. By the inscribed angle theorem, angle ABC is equal to angle AOC, and angle ADC is equal to angle AOD. Therefore, we have:

x + y = angle ABD + angle BCD = angle AOC + angle AOD = angle AOC + (180 – angle AOC) = 180 degrees

Thus, we have shown that the opposite angles of a cyclic quadrilateral are supplementary.

5. What are some examples of real-world applications of cyclic quadrilaterals?

A: Cyclic quadrilaterals are used in a variety of fields, including engineering, architecture, and physics. For example, the design of the circular gears used in many mechanical systems is based on the properties of cyclic quadrilaterals. In architecture, the shape of many domes and arches is based on the geometry of cyclic quadrilaterals.

Introduction

The volume of a pyramid is the space engulfed between the faces of the pyramid. The volume of the pyramid depends on the area of the base of the pyramid and the height of the pyramid. The different types of the pyramid consist of different bases so the volume of the pyramid differs from each other as the areas of the bases are different because a triangular pyramid has a triangle base and a square pyramid has a square base, the base areas are different so, the volume of the pyramid also different.

What is a Pyramid?

A pyramid is a three-dimensional solid structure with a polygon base and the other faces as triangles with all the vertex of the base linked to a single point called the apex of the pyramid. All the triangle faces are lateral faces. There are many types of pyramids each differing by the different-sided polygon base. If the base of the pyramid has n-sides then the pyramid has n+1 faces, n+1 vertices, and 2n edges. The perpendicular distance from the apex to the base is called the height of the pyramid, the perpendicular distance from the apex to any edge of the base is called the slant height of the pyramid.

Example: The pyramids of Egypt, etc.

Types of pyramids

The polygon bases make the pyramids differ from each other. Each type of pyramid has a different polygon base.

Triangular Pyramid: If the base of the pyramid is a triangle then the pyramid is called a triangular pyramid.  A triangular base is a three-sided polygon.

Number of faces of the pyramid (n=3) : 3 + 1 = 4

Number of vertices of the pyramid (n=3) : 3 + 1 = 4

Number of edges of the pyramid (n=3) : 23 = 6

Square Pyramid: If the base of the pyramid is a square then the pyramid is called a square pyramid.  A square base is a four-sided polygon. All sides of the base are equal in length.

Number of faces of the pyramid (n=4) : 4 + 1 = 5

Number of vertices of the pyramid (n=4) : 4 + 1 = 5

Number of edges of the pyramid (n=4) : \(2 \times 4 = 8\)

Rectangular Pyramid: If the base of the pyramid is a rectangle then the pyramid is called a rectangular pyramid. A rectangular base is a four-sided polygon. All the opposite sides of the base are equal in length.

Number of faces of the pyramid (n=4) : 4 + 1 = 5

Number of vertices of the pyramid (n=4) : 4 + 1 = 5

Number of edges of the pyramid (n=4) : \(2 \times 4 = 8\)

Hexagonal Pyramid: If the base of the pyramid is a hexagon then the pyramid is called a hexagonal pyramid. A hexagon base is a six-sided polygon.

Number of faces of the pyramid (n=6) : 6 + 1 = 7

Number of vertices of the pyramid (n=6) : 6 + 1 = 7

Number of edges of the pyramid (n=4) : \(2 \times 6 = 12\)

There are other types of pyramids like a pentagonal pyramid with a five-sided pentagon as the base, etc…

These pyramids also differ as regular and irregular pyramids based on the base, if the base of the pyramid is regular then it is a regular pyramid. If the base of the pyramid is irregular then it is an irregular pyramid.

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What is the volume of a Pyramid?

The volume of a pyramid depends upon the base area and the height of the pyramid (the perpendicular distance from the apex point to the base of the pyramid).

Consider the area of the base of the pyramid as B.

The height of the pyramid is H.

The formula for the volume of the pyramid:

The volume of the pyramid = \(\frac{1}{3} \times B \times H\)

Units of the volume of the pyramid would be cubic units. Also should convert all the parameter units into one unit before calculating the volume of the pyramid.

As we know there are different types of pyramids based on different bases (different bases contain different base areas) let’s check the volume of the pyramid for the different types of the pyramid.

Triangular Pyramid: We know that a triangular pyramid contains a triangle base. The volume of a triangular pyramid would be:

Consider H as the height of the pyramid, h as the height of the base triangle for the base that has length b:

The area of the base triangle would be B= \(\frac{1}{2} \times b \times h\)

Substituting B in the volume of the pyramid formula:

The volume of the triangular pyramid =\( = \frac{1}{3} \times B \times H = \frac{1}{3} \times \frac{1}{2} \times b \times h \times H\)

The volume of the triangular pyramid =\(\frac{1}{6} \times b \times h \times H\)             

Square Pyramid:  We know that a square pyramid contains a square base. The volume of a square pyramid would be:

Consider H as the height of the pyramid, and a as the length of the side of the square:

The area of the base square would be B=\({a^2}\)

Substituting B in the volume of the pyramid formula:

The volume of the square pyramid\( = \frac{1}{3} \times B \times H = \frac{1}{3} \times {a^2} \times H\)

The volume of the square pyramid\( = \frac{1}{3} \times {a^2} \times H\)

Rectangular pyramid:  We know that a rectangular pyramid contains a rectangle base. The volume of a rectangular pyramid would be:

Consider H as the height of the pyramid, l as the length of the rectangle base, and b as the breadth of the rectangle base,

The area of the base rectangle would be \(B = l \times b\)

Substituting B in the volume of the pyramid formula:

The volume of the rectangular pyramid\( = \frac{1}{3} \times B \times H = \frac{1}{3} \times l \times b \times h\)

The volume of the rectangular pyramid\( = \frac{1}{3} \times l \times b \times h\)

Hexagonal (regular) Pyramid: We know that a hexagonal pyramid contains a hexagon base. Considering this is a regular hexagon all the side lengths of the hexagon would be the same. The volume of a hexagonal pyramid would be:

Consider H as the height of the pyramid, and a as the length of the side of the hexagon

Solved Examples

1. A square pyramid has a height of 10cm and the length of the side of the square base is 3cm. Find the volume of the square pyramid?

Given the height of the pyramid H = 10cm, 

The length of the side of the square base a = 3cm

2. A rectangular pyramid has a height of 15cm, the length of the rectangular base is 5cm, and the breadth of the rectangular base is 4cm. Find the volume of the rectangular pyramid?

Word Problems

1. Consider an Egyptian pyramid with a square base of 50m side length and height of the pyramid is 150m, Find the volume of the Egyptian pyramid?

Given the height of the pyramid H = 150m, 

The length of the side of the square base a = 50m

Summary

A pyramid is a polyhedron as the faces of a pyramid is made up of different polygons. The volume of a pyramid is the space occupied by the pyramid in the three-dimensional space. In this tutorial, we learned about pyramids, different types of pyramids, the volume of a pyramid, the volume of different types of pyramids, how to calculate the volume of the different types of a pyramid, and a few examples .

Frequently Asked Questions

1. What is the volume of a triangular pyramid?

The volume of the triangular pyramid = \(\frac{1}{6} \times b \times h \times H\)

H is the height of the pyramid, b and h are the base and height of the base triangle.

2. What is meant by an irregular hexagonal pyramid?

An irregular hexagonal pyramid has an irregular hexagon (all side lengths are not equal)  as the base of the pyramid.

3. What is meant by slant height in a pyramid?

The perpendicular distance from the apex to any edge of the base is called the slant height of the pyramid.

4. What is the volume of a regular hexagonal pyramid?

The volume of the hexagonal pyramid\( = \frac{{\sqrt 3 }}{2} \times {a^2} \times H\)

H is the height of the pyramid, a is the side length of the hexagonal base.

5. What is the volume of a rectangular pyramid?

The volume of the rectangular pyramid\( = \frac{1}{3} \times l \times b \times h\)

H is the height of the pyramid, l and b are the length and breadth of the base rectangle.

Also Read : Volume of Cuboid
Volume Of Sphere

Introduction

The frustum is made up of a combination of solids. The volume of frustum means the space required by the frustum. In the previous tutorial, we have seen that the volume of a combined solid is equal to the summation of the volume of individual solids. In this tutorial, we will learn the basic structure, properties, and formula to determine the frustum volume with solved examples.

What is Frustum?

The word “frustum” originated from the Latin word “morsel”.  The frustum is a portion of a pyramid or a cone, which lies between one or two parallel planes cutting this. Consider a cone cut into two parts by a plane (parallel to the base). The base faces look like polygonal shapes, and the side faces look like trapezoidal shapes. The portion between the baseline and the parallel plain is known as a frustum (as shown in the figure). It is also known as a truncated cone. If the side lengths of the frustum are of equal length, then it is known as a uniform prism. The real-life examples of the frustum include drinking glasses, pyramids, buckets, some space capsules, etc. 

What is the Volume of a Frustum?

The volume of the frustum is defined as the space occupied by the object (frustum). The unit of volume of a frustum is generally expressed in \(c{m^3}{\mkern 1mu} {\rm{,}}{m^3}{\mkern 1mu} or{\rm{,}}i{n^3}\). The volume of any frustum can be determined if the height and the area of the two bases are known to us. Hence, the following formula can be used to evaluate the volume of a frustum of any shape.

where V and H are the volume and height of any frustum, respectively. In addition, \({A_1}\ and  \({A_2}\ refer to the area of two bases of the frustum.

where R = Radius of the lower base of the frustum

            r = Radius of the upper base of the frustum 

            h = Distance from upper base to the apex of the cone 

Let’s derive the above formula by two methods.

Method 1:

We can observe that the bases (lower and upper) of the frustum are of circular shapes.

Method 2:

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Solved Examples

Example 1:

The side length of the bases of a frustum are 9 cm and 4 cm. Both bases are square. The height of the frustum is 6 cm. Find the volume of the frustum.

Solution:

Example 2:

The height and radius of the large base of a frustum are 15 cm and 5 cm, respectively. The radius of the small base is 2.5 cm. Evaluate the volume of the frustum.

Solution:

Summary

The present tutorial gives a brief introduction about the volume of a frustum. The basic definition of the frustum and some real-life examples are illustrated. In addition, the formulae and their derivation to determine the volume of a frustum have been depicted in this tutorial. Moreover, some solved examples have been provided for better clarity of this concept. In conclusion, the present tutorial may be useful for understanding the basic concept of the volume of a frustum.

Frequently Asked Questions

1. What do you mean by the frustum of a cone?

If a cone is cut into two parts by a horizontal plane, then the lower part of the cone (opposite to the apex) is known as the frustum of a cone.

2. What is the formula to determine the curved surface area of the frustum?

The curved surface area of the frustum can be determined by using the following formula. 

3. What are the possible shapes of the base of the frustum?

The base of the frustum can be of any shape. They may be of square, rectangular, or circular shape.

4. Give some examples of the frustum.

Some real-life examples of frustum include table lamps, buckets, glasses, glass tumblers, etc.

5. What are the properties of a frustum?

The properties of a frustum are summarized below.

• The height of a frustum is the perpendicular distance between the two bases.
• Each element of a frustum of a cone is similar to the original cone. 

Also Read:
Volume of Cube
Volume of a Pyramid

Introduction

There are many geometrical solids around us. For example, two kids are playing board games using dice & coins. One boy is trying to solve Rubik’s cube. One thing that is common in the above examples is that these objects are in the shape of cubes. In this tutorial, we are going to discuss the topic of cubes & volume of a cube. A cube is a three-dimensional solid object having equal sides & faces. A cube has six faces, twelve edges & vertices. It is also known as the regular hexahedron or square prism. Volume of a cube is the amount of space occupied by the cube. It is one of the essential & fundamental concepts in geometry. The concept of the volume of a cube is majorly used for finding the capacity of a cubical tank.

What is a cube?

A cube is a symmetrical three-dimensional solid object bounded by six faces, facets or sides with three meeting at the point or corner known as the vertex. The cube has six faces, twelve edges & eight vertices. The dimensions of a cube are the same. It is also known as the regular hexahedron or square prism. Cube is a platonic solid (solid having congruent faces.)

The above figure represents a cube. Dice, ice cubes & Rubik’s cubes are some real-life examples of cubes. 

Some properties of the cube :

• A cube has 12 edges, 6 faces & 8 vertices. Faces of the cubes are square, therefore length, breadth & height of the cubes are equal.
• The angles between any edges & faces are \({90^0}\).
• The opposite planes or faces are parallel to each other.
• Three edges & three planes meet at each vertex of the cube.

What is volume of cube ?

The volume of a cube is a three-dimensional space occupied by the cube. For computing, the volume of the cube has two different formulas depending upon different parameters. By knowing the length of an edge of the cube we can calculate the volume of a cube. Also, the volume of a cube can be calculated by using a length of a diagonal. the volume of a cube is expressed in cubic units. Most of the time volume of cubes is expressed in SI units  \({m^3}\) in CGS units, \({cm^3}\) & litre.

Volume of a cube can be calculated by using two methods :

1) By using edge length 

2) By using diagonal

1) By using edge length

The volume of a cube having edge length ‘l’ can be calculated as 

Volume of a cube = length x breadth x height

 \(\begin{array}{l} = l \times l \times l\\ = {a^3}\end{array}\)

Derivation for a volume of a cube:

Consider a square sheet. The area of a sheet will be taken as surface area, the area of a sheet is length x breadth.

As the sheet is square. It has equal length & breadth, therefore the surface area will be \({a^2}\).

By stacking multiple sheets on top of each other square can be formed so that we will get height ‘a’ of a cube. 

Now, we can conclude as the overall area covered by the cube will be the area of the base multiplied by the height.

So, the volume of a cube \( = {a^2} \times a = {a^3}\)

So here we can conclude as the Volume of the cube \( = {\left( {side} \right)^3} \)

For example, the Volume of a cube having a side of 3 m can be calculated as  

             \( = {\left( {side} \right)^3} = {\left( 3 \right)^3} = 27{m^3}\)

2) By using diagonal 

Volume of a cube can be calculated by using a formula,

Volume of a cube \( = \frac{{\sqrt 3  \times {d^3}}}{9}\)

Here d is the diagonal of the cube 

For example, Length of a diagonal is 3 cm, then the volume of the cube will be

\[ = \frac{{\sqrt 3  \times {d^3}}}{9} = \frac{{\sqrt 3  \times {{\left( 3 \right)}^3}}}{9} = 3\sqrt 3 c{m^3}\]

Some other important formulae of a cube:

i)Total surface area of the cube =\(6{l^2}\) units

ii)Lateral surface area of the cube =\(4{l^2}\) units

Also Read: how to calculate volume of cuboid
                  how to find the volume of a sphere

Solved examples:

Q 1) Calculate the volume of a cube if the edge length of a cube is 

i) 9 cm

ii) 5.2 cm

 iii) 7.5 cm

   ii) 5.2 cm

 iii) 7.5 cm

Q 2) Calculate the edge length of a cube if the volume of the cube is

Q 3) Compute the volume of a cube which has a total surface area of 661.5 sq. cm.

Solution: Here, the total surface area = 661.5 sq. cm and the volume of a cube =?

Using the formula for the total surface area of a cube,

Q 4) Calculate the volume of the cube having a diagonal of 5 units.

Solution: Here length of a diagonal = 5 units 

Volume of a cube by a diagonal formula is given as,

Therefore, the volume of the cube is 24.05 cubic units

Word problems :

1) A cubical tank can store 1331000 ml of water. Then compute the side of the tank in cm.

Solution: Volume of a tank = 1331000 ml

2) Calculate the number of cubes when we cut a cube with an edge of 27 cm into cubes having an edge length of 3 cm?

Summary :

In this tutorial, we have learned about cubes & how to calculate the volume of cubes. A cube is a three-dimensional solid object having equal faces & edge lengths. Volume of a cube is a space occupied by a cube. Rubik’s cube, cubical tank, cubical box, and dice are some real-life examples of cubes. The volume of a cube can be calculated by two methods. The first is by using the edge length formula & by using the diagonal formula. This concept has wide application in real life. The concept of the volume of a cube is mainly used to determine the capacity of a cubical tank & find the side of a tank. This tutorial will surely help you to understand cubes & volume of cubes.

 

Frequently Asked Questions 

1. Why cube is known as a regular hexahedron?

Ans. A regular hexahedron is a 3D solid object having six congruent faces. Cube has six congruent faces, therefore cube is known as the regular hexahedron.

2. State the difference between cube & cuboid?

Ans. A cube is a three-dimensional solid object having all square faces whereas a cuboid is a three-dimensional object having all rectangle faces.

3. Can a prism be a cube?

Ans. A cube is a prism because a cube is considered one of the platonic solids.

4. State the difference between the surface area & lateral surface area of a cube?

Ans. For the calculating surface area of a cube sum of the area of all faces is taken whereas for calculating lateral surface area sum of only four surfaces is taken.

i)Total surface area of the cube = \(6{l^2}\) units

ii)Lateral surface area of the cube =  \(4{l^2}\) units

5. Explain what is net of a cube is?

Ans. The net of a cube is formed when the square faces of the cube are flattened by separating at the edges to form a 2D figure. Through that figure, we can see six faces of the cube.

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Introduction

The simplest form in which a number can be written is called standard form. The goal is to simplify number calculations as well as reading and writing. Every mathematical concept, including integers, fractions, equations, and expressions, has a specified standard form. The standard form can be thought of as the most basic approach to represent a mathematical element. Any value between \(1.0\) and \(10.0\) that can be expressed as a decimal number and multiplied by a power of 10 is referred to as being in standard form.

Expressing large numbers in the standard form

The easiest form of decimal numbers to read and write is in standard form. For instance, \(9 \times {10^{ – 3}}\) is the standard form of the decimal number \(0.009\). Very large or very small numbers might be challenging to read or write at times. We therefore use standard form while writing them. Any number can be written in standard form, not just decimals. Some fractions result in decimal numbers with additional digits at the thousandth, hundredth, or tenth places.

In general, we can say that it is the standard form representation of rational numbers. The definition of a rational number is any number that can be written as p/q, where p and q are both integers. As an example, \(1/13,{\rm{ }}4/15,{\rm{ }}8/9\) , etc.

Standard form

In mathematics, the most usual way to represent a specific element is called standard form. Every mathematical concept, including big numbers, small numbers, equations, and lines, has a standard form. Explore this exciting idea of standard form as it relates to many math concepts, including fractions, equations, algebra, slope, and learning the standard from a formula.

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What are standard form

A standard form is a way to express a particular mathematical idea, such as an equation, number, or expression, in prose that follows to a set of rules. The standard form is used to express very big or very small numbers clearly. For instance, \(4,500,000,000\) years is how \(4.5\) billion years is written. As you can see, it is difficult and time-consuming to write a large number like \(4.5\) billion in its number form. There is also a chance that we may write a few of more or less than necessary. It is quite beneficial in this situation to write the number in standard form. For instance, \(4,500,000,000\) in standard form equals  \(4.5 \times {10^9}\) . Additionally to integers, other mathematical constructs like fractions, equations, expressions, and polynomials can.

Standard form of a number

Writing a very large or extremely small number using powers of \(10\)  multiplied by values between  \(1\)  and \(10\) is known as scientific notation. 3890, for instance, can be written as \(3.89 \times {10^3}\) . Use positive powers of \(10\)  to express these values, which are bigger than \(1\) . The negative power of ten is used for numbers lower than one. For instance, \(0.0451\) may be expressed as \(4.51 \times {10^{ – 2}}\).

Express the following numbers in standard form

(1.) \(0.0000000000069\)

\( = \frac{{69}}{{10000000000000}}\)

\( = \frac{{69}}{{{{10}^{13}}}} = 69 \times {10^{ – 13}}\)

\( = 6.9 \times 10 \times {10^{ – 13}}\)

\( = 6.9 \times {10^{ – 12}}\)

(2.) \(90000000\)

\(\; = 9 \times {10^7}\)

(3.)  \(2650000000\)

\(\; = 2.65 \times {10^9}\)

How to write numbers in standard form 

The stages to writing a number in its standard form are as follows:

Step 1: Write the first digit of the supplied number in step one.

Step 2: After the first number, add the decimal point.

Step 3: Next, count how many digits there are in the supplied number after the first one and express that number as a power of \(10\) .

For instance, the number is \(52300000000\) . Thus, the following is how a number is represented in standard form:

The initial number is \(5\) .

Step 2: Input the decimal point to make \(5\) into “\(5\)”.

Step 3: There are \(10\) digits after the number \(5\) .

Consequently, \(5.23 \times {10^{10}}\) is the conventional form of \(52300000000\) .

Express the following numbers in usual form

Shift the decimal to the left by the number of places equal to the power of  \(10\) to convert a smaller number (negative powers of \(10\) ) to its standard form. The decimal point must be moved to the right by the number of places equal to the power of 10 in order to change a large number (positive powers of \(10\) ) to its basic format.

4. \(4.5{\rm{ }} \times {10^5} = {\rm{ }}450000\)
5. \(7.8 \times {10^6} = 7800000\)
6. \(9.7 \times 109 = 9700000000\)

Standard form definition 

A standard form is a way to express mathematical ideas like an equation, an expression, or some numbers. Example 2,500,000,000 years is the same as \(2.5\) billion years. As you can see, it is challenging and time-consuming to read or write a large number like \(2.5\) billion. Therefore, we utilize the standard form to precisely write large or tiny integers.

Solved examples 

Example 1: Use an exponential function to represent the separation between the Earth and the Sun.

Answer: The Earth’s distance from the Sun is \(1496000000km\) .

Therefore,

\(1496000000Km{\rm{ }} = 1.496 \times 109Km\)

Example 2: Use the conventional form to describe the size of blood cells.

Answer: Human blood cells typically measure \(0.000015\) m in size.

Therefore,

\(0.000015 = 1.5 \times {10^{ – 5}}m\)

Example 3.Write \(3253\) in standard form.

Answer: \(3.253 \times 1000\) can be used to represent the number 3253. The standard form of 3253 is \(3.253 \times {10^3}\) .

Conclusion

Technically speaking, large numbers are defined as those that are greater than those observed frequently. Large numbers in the number system are those that are typically bigger or greater than the other numbers. For instance, large figures like 1 lakh, 1 million, 1 billion, etc. are not ones we use frequently. In essence, standard form is used to indicate these large integers.

Frequently Asked Questions

1. What is the canonical form of a number?

Ans. The canonical form of a number is a way of writing numbers in a form

That follows certain rules. Numbers that can be written as decimal numbers

Between 1.0 and 10.0 multiplied by powers of 10 are called canonical forms.

2. What is the standard form of a decimal number?

Ans. The standard form of a decimal number is the representation of a given Decimal number raised to the tenth power in order to simplify to the original Value.

3. Are the standard form and the standard notation of decimal numbers the Same?

Ans. The standard form is also called the standard notation. So, both are the

Same.