# Area of a Rectangle - Examples, Exercises and Solutions

## How Do You Calculate the Area of a Rectangle?

Compared to other geometric figures, the rectangle is one of the simplest to work with.

One of the most frequent questions that comes up in exams is related to how to calculate the area of the rectangle.

Before we focus on it, let's do a brief review.

### The formula for calculating the area of a rectangle is as follows:

$Base~DC\times Height~AD=Area~of~rectangle$

## examples with solutions for area of a rectangle

### Exercise #1

Look at the rectangle ABCD below.

Side AB is 6 cm long and side BC is 4 cm long.

What is the area of the rectangle?

### Step-by-Step Solution

Remember that the formula for the area of a rectangle is width times height

We are given that the width of the rectangle is 6

and that the length of the rectangle is 4

Therefore we calculate:

6*4=24

24 cm²

### Exercise #2

Given the following rectangle:

Find the area of the rectangle.

### Step-by-Step Solution

We will use the formula to calculate the area of a rectangle: length times width

$9\times6=54$

54

### Exercise #3

The width of a rectangle is equal to $4^2$ cm and its length is $5$ cm.

Calculate the area of the rectangle.

### Step-by-Step Solution

We use the formula to calculate a rectangle: length times width:

According to the existing data:

$4^2\times5^=S$

$S=16\times5$

$S=80$

80

### Exercise #4

The area of the rectangle below is equal to 63.

AC = 7

How long is side AB?

### Step-by-Step Solution

We use the formula to calculate a rectangle: length times width:

$AB\times AC=S$

We place the existing data into the formula:

$AB\times7=63$

$7AB=63$

We divide both sides by 7:

$AB=9$

9

### Exercise #5

The area of a rectangle is equal to 8.

Calculate the perimeter of the rectangle.

### Step-by-Step Solution

According to the properties of the rectangle, all pairs of opposite sides are equal.

$AB=CD=8$

$AC=BD=2$

Now we calculate the perimeter of the rectangle by adding all the sides:

$4+4+2+2=8+4=12$

12

## examples with solutions for area of a rectangle

### Exercise #1

The area of the rectangle below is equal to 24.

Calculate the perimeter of the rectangle.

### Step-by-Step Solution

Given that in a rectangle all pairs of opposite sides are equal to each other, it can be argued that:

$AB=CD=6$

$AC=BD=4$

Now calculate the perimeter of the rectangle by adding all the sides:

$4+4+6+6=$

$8+12=20$

In other words, the data of the rectangle's area is unnecessary, since we already have all the data to calculate the perimeter, and we do not need to calculate the other sides.

20

### Exercise #2

Below is the rectangle ABCD.

It has an area of 42 cm² and side AD is equal to 12 cm.

What is the length of side DC?

### Step-by-Step Solution

Remember that to calculate the area of the rectangle, we multiply the length by the width.

Therefore:

$42=12\times CD$

$42=12CD$

Divide both sides by 12:

$3.5=CD$

3.5

### Exercise #3

The perimeter of the rectangle below is equal to 30.

What is the area of the rectangle?

### Step-by-Step Solution

We use the formula to calculate the area of a rectangle: length times width:

$AC\times AB=S$

We replace the existing data:

$5\times10=50$

That is, the information that the perimeter of the rectangle is equal to 30 is unnecessary, since all the data to calculate the area already exist and it is not necessary to calculate the other sides.

50

### Exercise #4

Look at the given rectangle made of two squares below:

What is its area?

### Step-by-Step Solution

In a square all sides are equal, therefore we know that:

$AB=BC=CD=DE=EF=FA=5$

The area of the rectangle can be found in two ways:

1. Find one of the sides (for example AC)

$AC=AB+BC$

$AC=5+5=10$

and multiply it by one of the adjacent sides to it (CD/FA, which we already verified is equal to 5)

$5\times10=50$

2. Find the area of the two squares and add them.

The area of square BCDE is equal to the multiplication of two adjacent sides, both equal to 5.

$5\times5=25$

Square BCDE is equal to square ABFE, because their sides are equal and they are congruent.

Therefore, the sum of the two squares is equal to:

$25+25=50$

50

### Exercise #5

Look at the two rectangles in the figure:

What is the area of the white area?

### Step-by-Step Solution

As we know that EFGD is a rectangle, we also know that DE is equal to 2 and DG is equal to 4

In a rectangle, each pair of opposite sides are equal and parallel, therefore:

$ED=FG=2$

$DG=EF=4$

Now we calculate the area of the orange rectangle EFGD by multiplying the length by the width:

$2\times4=8$

Now we calculate the total area of the white rectangle ABCD:

$AD=AE+ED=2+2=4$

$DC=DG+GC=4+5=9$

The area of the entire rectangle ABCD is:

$4\times9=36$

Now to find the area of the white part that is not covered by the area of the orange rectangle, we will subtract the area of the rectangle EFGD from the rectangle ABCD:

$36-8=28$

28 cm²

## examples with solutions for area of a rectangle

### Exercise #1

The rectangle ABCD is shown below.

$BD=25,BC=7$

Calculate the area of the rectangle.

### Step-by-Step Solution

To find side DC we will use the Pythagorean theorem:

$(BC)^2+(DC)^2=(DB)^2$

Now we will replace the existing data in the theorem:

$7^2+(DC)^2=25^2$

$49+DC^2=625$

$DC^2=625-49=576$

We extract the root:

$DC=\sqrt{576}=24$

168

### Exercise #2

The area of a rectangle is 256 cm².

One side is 4 times longer than the other.

What are the dimensions of the rectangle?

### Step-by-Step Solution

To find the area of the rectangle, we multiply the length by the width.

According to the data in the statement, one side will be equal to X and the other side will be equal to 4X

Now we replace the existing data:

$S=x\times4x$

$256=4x^2$

We divide the two sections by 4:

$64=x^2$

We extract the square root:

$x=\sqrt{64}=8$

If we said that one side is equal to x and the other side is equal to 4x and we know that x=8

From here we can conclude that the sides of the rectangle are equal:

$8,8\times4=8,32$

8 x 32

### Exercise #3

Calculate the area of the rectangle below using the distributive property.

### Step-by-Step Solution

The area of the rectangle is equal to the length multiplied by the width.

We write the exercise according to the existing data:

$7\times(4+5)$

We solve the exercise using the distributive property, that is, we multiply 7 by each of the terms in parentheses:

$(7\times4)+(7\times5)=$

We solve the exercise in parentheses and obtain:

$28+35=63$

63

### Exercise #4

The width of a rectangle is equal to$x$ cm and its length is $x-4$ cm.

Calculate the area of the rectangle.

### Step-by-Step Solution

The area of the rectangle is equal to the length times the width:

$S=x\times(x-4)$

$S=x^2-4x$

$X^2-4X$

### Exercise #5

Calculate the area of the rectangle

### Step-by-Step Solution

Remember that the formula to calculate the area of a rectangle: width X length

$S=w⋅h$

Where:

S = area

w = width

h = height

We take data from the sides of the rectangle in the figure.

$w=x+5$$h=y+2$

Now we replace in the formula to calculate the area of the rectangle:

$S=w⋅h=(x+5)(y+2)$

We use the formula of the extended distributive property:

$(a+b)(c+d)=ac+ad+bc+bd$

We replace and solve:

$S=(x+5)(y+2)=(x)(y)+(x)(2)+(5)(y)+(5)(2)$

$(x)(y)+(x)(2)+(5)(y)+(5)(2)=xy+2x+5y+10$

Therefore, the correct answer is option C: xy+2x+5y+10.

$xy+2x+5y+10$