A rectangle is a two-dimensional figure with four sides, four vertices, and four right angles. One of the main characteristics of the rectangle is that its two opposite sides are equal and parallel to each other. The area of a rectangle is defined as the space covered by the 2D figure. Alternatively, we can also consider the area as the space within the perimeter of the rectangle.
Here, we will see why the area of a rectangle is the product of its two sides. In addition, we will look at several exercises of areas of a rectangle to master the use of the formula.
What is the formula to find the area of a rectangle?
The formula for calculating the area of a rectangle is A=Base×Height. The area of a rectangle depends on its sides. Basically, the formula for the area is equal to the product of the base and the height of the rectangle.
On the other hand, when we talk about the perimeter of the rectangle, we know that it is equal to the sum of its four sides. Therefore, we can say that the region enclosed by the perimeter of the rectangle is its area.
The square is a special case of a rectangle, which has all equal sides. Since all the sides in a square are equal, the area of the square will be equal to the length of one of its sides squared.
| Area of rectangle = Base × Height $latex A=b\times a$ |
The area of any rectangle is calculated once its base and height are known. By multiplying the base times the height, the area of the rectangle gets dimensions in square units.
How to calculate the area of a rectangle?
We can find the area of a rectangle by following these steps:
Step 1: Identify the dimensions of the base and the height from the given information.
Step 2: Multiply the base and height values.
Step 3: Write the answer in square units.
Why is the area of a rectangle equal to base times height?
In the rectangle below, its diagonal divides it into two equal right triangles. Therefore, the area of the rectangle will be equal to the sum of the areas of these two triangles.
We have the rectangle ABCD:
The diagonal AC divides the rectangle into two right triangles, $latex \Delta$ABC and $latex \Delta$ADC.
We know that $latex \Delta$ABC and $latex \Delta$ ADC are congruent triangles, so we have:
⇒ Area (ABCD) = Area (ABC) + Area (ADC)
⇒ Area (ABCD) = 2 × Area (ABC)
⇒ Area (ABC) = $latex \frac{1}{2}$ × base × height
⇒ Area (ABCD) = 2 × ($latex \frac{1}{2}$ × b × h)
⇒ Area (ABCD) = b × h
How to calculate the sides of a rectangle knowing the area?
If we know the area of a rectangle, we can calculate its sides using the formula for the area of a rectangle. The formula for the area of a rectangle is A = base × height.
In the case of a square, this becomes very easy since a square is a special case of a rectangle that has all its sides equal, so its area is equal to one of its sides squared. Therefore, if we have the area of a square, we can calculate its sides by taking the square root of the area.
For example, if we have that the area of a square is 100 square meters, we can take the square root to obtain 10 meters, which is the length of its sides.
In the case of a rectangle with different side lengths, this is a bit more complicated since we need to know the area and one of its sides to calculate the length of the other side.
For example, if the area of a rectangle equals 100 square meters and one of its sides is 20 meters, we simply divide 100 by 20 to get the length of the other side. Thus, the length of the other side is 5 meters
Alternatively, we can find the length of the sides of a rectangle if we know its area and the proportions of its sides. For example, if we have that the area of a rectangle is equal to 100 square meters and the proportions of its sides are 5:20, we know that one of its sides is 5 meters and the other is 20 meters.
Area of a rectangle – Examples with answers
The following examples use the formula for the area of a rectangle to get the answer. Each example has its respective solution, but it is recommended that you try to solve the exercises yourself before looking at the answer.
EXAMPLE 1
Find the area of a rectangle that has a base of 20 cm and a height of 12 cm.
Solution
We have the following data:
Base = 20 cm
Height = 12 cm
We have the formula A = base × height. Using the given information, we have:
$latex 20 \times 12 = 240$
Therefore, the area is 240 cm².
EXAMPLE 2
Find the area of a blackboard with dimensions 140 cm in base and 90 cm in height.
Solution
We have the following information:
Base = 140 cm
Height = 90 cm
Now, we use the formula A = base × height with the information given:
$latex 140 \times 90 = 12600$
Therefore the area of the board is 12 600 cm².
EXAMPLE 3
The base of a rectangle is 20 cm and its area is 120 cm². What is the length of its height?
Solution
We have the following information:
Base = 20 cm
Area = 120 cm²
We can use the formula A = base × height. Thus, we substitute the information to get:
base $latex = \frac{120}{20}= 6$
Therefore, the height is 6 cm.
EXAMPLE 4
A square has sides with a length of 8 m. What is its area?
Solution
We have the following information:
Side = 8 m
We know that a square is a rectangle that has equal sides, so its area can be found using A = side². Thus, we use the information to get:
$latex A={{8}^2}= 64$
Therefore, the area of the square is 64 m².
EXAMPLE 5
The area of a square is equal to 8100 cm². What is the length of one of its sides?
Solution
We have the following information:
Area = 8100 cm²
We know that a square is a rectangle that has equal sides, so we can use the formula A = side². Thus, we substitute to get:
side $latex = \sqrt{8100}=90$
Therefore, the length of one of the sides is 90 cm.
Area of a rectangle – Practice problems
Test what you have learned about the area of a rectangle with the following problems.
See also
Interested in learning more about rectangles? Take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
