Area of a Rectangle

• Introduction
• Understanding the Area of a Rectangle
• Formula of Area of Rectangle
• Steps to Calculate Area of Rectangle
• Deriving the Formula of Area of Rectangle 
• Understanding the Unit of Rectangle Area
• Finding the Area of a Rectangle Using Diagonal
• Finding the Area of a Rectangle Using Perimeter
• Solved Examples
• Practice Questions
• Frequently Asked Questions

Introduction 

A rectangle is a two-dimensional polygon with four sides, `4` vertices and `4` right angles. The opposite sides of a rectangle are equal and parallel to each other. The area of a rectangle refers to the space enclosed within its boundaries or the space within the perimeter of the rectangle. To find this area, you can use a specific formula and different methods based on the given dimensions. Let's explore how to calculate the area of a rectangle in this lesson.

Understanding the Area of a Rectangle

The area of a rectangle is the amount of space it takes up within its boundaries. You can think of it as the number of square units that fit inside the rectangle. This is measured using terms like square centimeters or square inches. For instance, think about the flat surface of a laptop screen or a chalkboard – those are rectangular shapes. To find the area of a rectangle, you use a formula that helps calculate the space it occupies. Another way to understand it is by looking at the space inside the boundaries of the rectangle, which is determined by its perimeter.

Formula of Area of Rectangle

The formula for finding the area of a rectangle helps determine the space it covers within its boundaries. You calculate it by multiplying its length and width. So, if a rectangle has a length of '`l`' and a width of '`w`', its area can be expressed:

Area of Rectangle, \( A = l \times w \)

This simply means that to find the area, you multiply the length by the width. To apply this formula correctly, ensure that the units for both length and width are the same. For instance, if the length is given in centimeters and the width in meters, convert both to either centimeters or meters to maintain consistency. This makes calculations accurate and easier to understand.

Steps to Calculate Area of Rectangle

The area of a rectangle is found by multiplying its length and width. The area of a rectangle can be calculated using the following steps:

Step `1`: Write down the length and width of the rectangle. Make sure both have the same unit.

Step `2`: Multiply the length by the width.

Step `3`: Express the answer in respective square units.

Let’s consider an example to understand the calculation of the area of a rectangle.

Example: Find the area of the rectangle whose length is \(25 \, \text{cm}\) and width is \(10 \, \text{cm}\). 

Solution: 

Formula of area of rectangle is: 

`\text{Area} = l \times w`

`\text{Area} = 25 \times 10`

`\text{Area} = 250 \ \text{cm}^2`

Thus, the area of the rectangle is \( 250 \, \text{cm}^2 \).

Deriving the Formula of Area of Rectangle 

In this section, we're going to explore why we calculate the area of a rectangle using the formula: \( A = l \times b \).

Imagine a rectangle named `ABCD` with a diagonal line `BD` running through it. This diagonal line splits the rectangle into two equal triangles. Since triangles \( ABD \) and \( BCD \) are congruent, we can represent the area of \( ABCD \) as twice the area of triangle \( BCD \). 

Let's focus on one triangle, `BCD`. The base of this triangle, `CD`, matches the length of the rectangle, let's call it \(l\), and `BC`, its height, matches the width of the rectangle, let's call it \(w\). Thus:

`\text{Area of Rectangle } ABCD = 2 \times \text{Area of Triangle } BCD`

`\text{Area of Rectangle } ABCD = 2 \times \left( \frac{1}{2} \times \text{Base} \times \text{Height} \right)`

`\text{Area of Rectangle } ABCD = 2 \times \frac{1}{2} \times CD \times BC`

`\text{Area of Rectangle } ABCD = CD \times BC`

`\text{Area of Rectangle } ABCD = l \times w`

Therefore, the area of the rectangle is equal to its length multiplied by its width.

Understanding the Unit of Rectangle Area

In geometry, when we talk about the size of a rectangle, we measure it in square units. Essentially, this means we're looking at how many squares of a certain size can fit inside the rectangle. Let's break this down with a simple example.

Imagine we have a rectangle with a length of \(6 \, \text{cm} \) and a width of \(3 \, \text{cm} \). Picture this rectangle divided into smaller squares, each side measuring \(1 \, \text{cm} \). These squares are what we call square cm. Now, if we count how many of these little squares can fit inside our rectangle and we find there are \(18 \) of them. 

So, we say the area of this rectangle is `18` `"cm"^2`. It's like we're covering the whole space of the rectangle with these square cm. And this concept applies no matter what units we use – whether it's inches, centimeters, or anything else, the idea remains the same.

Finding the Area of a Rectangle Using Diagonal

When you're given the diagonal and one side of a rectangle, you can figure out its area. The diagonal is the straight line connecting opposite corners inside the rectangle, and both diagonals are of equal length. There are two methods to find the area of a rectangle with this information.

Method `1`: 

To find the missing side of a rectangle, we can utilize Pythagoras' theorem. This helps us calculate the missing side of the rectangle using the length of the diagonal. Let's break it down with an example.

The diagonal of a rectangle is the line connecting opposite corners. We determine its length using Pythagoras’ theorem:

\( \text{Diagonal}^2 = \text{Length}^2 + \text{Width}^2 \)

We can then solve for the missing length or width of the rectangle:

\( \text{Length} = \sqrt{\text{Diagonal}^2 - \text{Width}^2} \)

OR

\( \text{Width} = \sqrt{\text{Diagonal}^2 - \text{Length}^2} \)

After obtaining the missing length, we use the formula for the area of a rectangle:

\( \text{Area} = \text{Length} \times \text{Width} \)

Substituting the length into the area formula, we get:

\( \text{Area} = \sqrt{\text{Diagonal}^2 - \text{Width}^2} \times \text{Width} \)

Or equivalently:

\( \text{Area} = \text{Width} \times \sqrt{\text{Diagonal}^2 - \text{width}^2} \)

Let's illustrate this with an example.

Example: Find the area of a rectangle whose length is `5` cm and whose diagonal is `13` cm.

Solution:

Length \(= 5 \, \text{cm} \)

Diagonal \(= 13 \, \text{cm} \)

First, we use Pythagoras' theorem to find the width of the rectangle.

\( \text{Diagonal}^2 = \text{Length}^2 + \text{Width}^2 \)

\( 13^2 = 5^2 + \text{Width}^2 \)

\( 169 = 25 + \text{Width}^2 \)

\( \text{Width}^2 = 169 - 25 \)

\( \text{Width}^2 = 144 \)

\( \text{Width} = \sqrt{144} \)

\( \text{Width} = 12 \, \text{cm} \)

Now that we have the width, we can use the formula for the area of a rectangle:

\( \text{Area} = \text{Length} \times \text{Width} \)
\( \text{Area} = 5 \times 12 \)
\( \text{Area} = 60 \, \text{cm}^2 \)

Therefore, the area of the rectangle is \(60 \, \text{cm}^2\). `d_1`

Method `2`:

If you have a rectangle and you know the lengths of both its diagonals, you can figure out its area easily. Let's say you have a rectangle named `ABCD`, and its diagonals are labeled as `AC` and `BD`.

If the length of `AC` is \(d_1\) and the length of `BD` is \(d_2\), then you can find the area of the rectangle using the formula:

\( \text{Area of rectangle ABCD} = \frac{1}{2} \times d_1 \times d_2 \)

This means that to find the area, multiply half of one diagonal by the other diagonal.

Let's illustrate this with an example.

Example: Find the area of a rectangle whose length of the diagonals is `20` cm and `26` cm.

Solution: 

\(d_1 = 20 \, \text{cm} \)

\(d_2 = 26 \, \text{cm} \)

The formula for area of rectangle is:

`\text{Area of rectangle} = \frac{1}{2} \times d_1 \times d_2`

`\text{Area of rectangle} = \frac{1}{2} \times 20 \times 26`

`\text{Area of rectangle} = 260 \ \text{cm}^2`

So, the area of the rectangle is \(260 \, \text{cm}^2\).

Finding the Area of a Rectangle Using Perimeter

If you know the perimeter and one side of a rectangle, you can find its area using the following steps: 

Step `1`: Write down the perimeter and the given dimensions.

Step `2`: Use the perimeter formula to determine the unknown side length.

Step `3`: Substitute the values of length and width into the formula of area of a rectangle.

Step `4`: Simplify the expression and add \( \text{unit}^2 \) to obtain the final answer.

Let's illustrate this with an example.

Example: Find the area of a rectangle if the perimeter is `30` cm and the length is `6` cm.

Solution: 

Perimeter \(= 30 \, \text{cm} \)

Length \(= 6 \, \text{cm} \)

Using the perimeter formula: 

Perimeter of rectangle \(= 2 \:(\text{l} + \text{w})\)

\( 30 = 2 \: (6 + \text{b}) \)

\(15 = 6 + \text{b} \)

\( \text{b} = 9 \, \text{cm} \)

So, the width of the rectangle is \(9 \, \text{cm} \)

Now,

`\text{Area of rectangle} = \text{l} \times \text{w}`

`= 6 \times 9`

`= 54 \ \text{cm}^2`

Thus, the area of the rectangle is \( 54 \, \text{cm}^2 \).

Solved Examples

Example `1`: A rectangular screen has a length of `13` in and a width of `7` in. Find its area.

Solution: 

Length \(= 13 \, \text{in} \)

Width \(= 7 \, \text{in} \)

`\text{Area of rectangle} = l \times w`

`= 13 \times 7`

`= 91 \ \text{in}^2`

So, the area of the rectangular screen is \(91 \, \text{in}^2 \).

Example `2`: The length of a rectangle is \( 8 \) cm and its area is \( 56 \) cm\(^2\). Find the width.

Solution: 

Given:

Length \(= 8 \, \text{in} \)

Area \(= 56 \, \text{cm}^2 \)

Area of rectangle \( = l \times w \), 

Therefore,

`b = \frac{A}{l}`

`b = \frac{56}{8}`

`b = 7`

So, the width of the rectangle is \( 7 \) cm.

Example `3`: The perimeter of a rectangular farm is \( 80 \) yards and its width is \( 10 \) yards. Find its length and area.

Solution: 

Given:  

Perimeter \(=80 \, \text{yards} \)

Width \(= 10 \, \text{yards} \)

Using the perimeter formula: 

Perimeter of rectangle \(= 2 \:(\text{l} + \text{w})\)

\( 80 = 2 \: (\text{l} + 10) \)

\(40 = \text{l} + 10 \)

\( \text{l} = 40 - 10 \)

\( \text{l} = 30 \, \text{yards} \)

So, the length of the rectangular farm is \(30 \, \text{yards} \)

Now,

`\text{Area of rectangle} = \text{l} \times \text{w}`

`= 30 \times 10`

`= 300 \ \text{square yards}`

Thus, the area of the rectangular farm is \( 300 \, \text{square yards} \).

Example `4`:  A rectangular swimming pool has a length of \( 20 \) feet and a width of \( 10 \) feet. If the cost of painting the floor of the pool is `$2` per square foot, what is the total cost of painting the pool floor

Solution: 

Given: 

Length of the swimming pool \(= 20 \, \text{ft} \)

Width of the swimming pool \(= 10 \, \text{ft} \)

`\text{Area of the swimming pool} = l \times w`

`= 20 \times 10`

`= 200`

So, the total area of the pool is \( 200 \) square feet.  

Now, to find the cost, we multiply the area by the cost per square foot:  

`\text{Total Cost} = 200 \times 2`

`= 400`

Therefore, it will cost \( $400 \) to paint the pool.

Example `5`: A wall whose length and width are \(50 \, \text{m} \) and \(30 \, \text{m}\) respectively requires painting. Determine the amount of paint needed if `1` litre of paint can cover \( 300 \, \text{m}^2 \) of the wall.

Solution: 

Given: 

Length of wall \(= 50 \, \text{m} \)

Width of wall \(= 30 \, \text{m} \)

`\text{Area of wall} = l \times w`

`= 50 \times 30`

`= 1500`

So, the total area of the wall is \( 1500 \, \text{m}^2 \).

Now, Paint required for \(300 \, \text{m}^2 \) of wall \( = 1 \, \text{litre} \)

`\text{Paint required for } 1500 \, \text{m}^2 \text{ of wall} = \frac{1}{300} \times 1500`

`= 5 \ \text{litres}`

Thus, `5` litres of paint is required to paint the wall.

Practice Problems

Q`1`. What is the area of a rectangle with length \( 8 \) units and width \( 5 \) units?

1. \( 13 \) sq units
2. \( 35 \) sq units
3. \( 40 \) sq units
4. \( 42 \) sq units

Answer: c

Q`2`. If the area of a rectangle is \( 72 \) sq units and its length is \(  9 \) units, what is its width?

1. \( 8 \) units
2. \( 6 \) units
3. \( 10 \) units
4. \( 12 \) units

Answer: a

Q`3`. Find the area of a rectangle whose length is twice its width and perimeter is `84` cm.

1. \( 288 \, \text{cm}^2 \)
2. \( 300 \, \text{cm}^2 \)
3. \( 392 \, \text{cm}^2 \)
4. \( 410 \, \text{cm}^2 \)

Answer: c

Q`4`. How many boxes of dimensions \( 15 \, \text{cm} \times 12 \, \text{cm} \) can be placed on a floor of dimensions \( 300 \, \text{cm} \times 120 \, \text{cm} \)?

1. \( 110 \)
2. \( 150 \)
3. \( 170 \)
4. \( 200 \)

Answer: d

Q`5`. A rectangular poster has dimensions \( 18 \times 12 \) inches. If the cost of printing is `$0.05` per square inch, what is the total cost of printing the poster?

1. `$6.00`
2. `$10.80`
3. `$21.60`
4. `$43.20`

Answer: b

Frequently Asked Questions

Q`1`. What is the formula for finding the area of a rectangle?

Answer: The formula for the area (\( A \)) of a rectangle with length (\( l \)) and width (\( w \)) is given by \( A = l \times w \).

Q`2`. What is the relationship between the area and perimeter of a rectangle?

Answer: There isn't a direct relationship between the area and perimeter of a rectangle. However, given the perimeter, one can find the dimensions of a rectangle and subsequently calculate its area.

Q`3`. If the length of a rectangle is doubled, how does it affect the area?

Answer: If the length of a rectangle is doubled (\( l \rightarrow 2l \)), the area becomes doubled as well. This is because the area of a rectangle is directly proportional to its length.

Q`4`. Can a rectangle have a negative area?

Answer: No, a rectangle cannot have a negative area. Area represents the extent of a surface, and it is always a non-negative quantity.

Q`5`.  If the width of a rectangle is `0`, what is its area?

Answer: If the width (\( w \)) of a rectangle is `0`, its area becomes `0` as well. This is because the area of a rectangle is determined by multiplying its length by its width.