The perimeter of a rectangle can be defined as the length of the outline of the rectangle. On the other hand, the area of the rectangle is a measure of the space occupied by the rectangle in two-dimensional space. We can calculate the perimeter of a rectangle using the formula p = 2(a+b) and we can calculate its area using the formula A = ab, where b is the base of the rectangle and a is its height.
In this article, we will learn about the perimeter and area of a rectangle in detail. We will learn about its formulas and apply them to solve some practice problems.
How to find the perimeter of a rectangle?
The perimeter of a rectangle can be calculated by adding the lengths of all its sides. In a rectangle, opposite sides are equal, so the perimeter of a rectangle is equal to twice the length of the base plus twice the length of the height of the rectangle.
| Perimeter = 2(Base + Height) $latex p = 2(a+b)$ |
where, a is the height of the triangle and b is the length of its base, as shown in the following diagram:
Proof of the formula for the perimeter of a rectangle
The perimeter of any geometric figure can be found by adding the lengths of all its sides. In the case of a rectangle, we have:
⇒ P = sum of its four sides
⇒ P = a + b + a + b (opposite sides are equal)
⇒ $latex P = 2(a + b)$
How to find the area of a rectangle?
The area of a rectangle can be calculated by multiplying the length of the base of the rectangle by its height. Therefore, we have the following formula:
| Area of rectangle = Base × Height $latex A=b\times a$ |
The area of a rectangle is expressed in square units. Therefore, we can find the area of a rectangle by following the steps below:
Step 1: Identify the lengths of the base and height of the rectangle. Make sure the dimensions are the same. If the base is given in inches, the height must also be in inches.
Step 2: Multiply the lengths of the base and height.
Step 3: Write the answer in square units..
Proof of the formula for the area of a rectangle
Let's use the following diagram to demonstrate the formula for the area of a rectangle:
We have the rectangle ABCD. Diagonal AC divides the rectangle into two right triangles, $latex \Delta$ABC and $latex \Delta$ADC. The area of the rectangle is equal to the sum of the areas of the two triangles.
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
Perimeter and area of a rectangle - Practice problems
EXAMPLE 1
What is the perimeter of a rectangle that has a base of 12 inches and a height of 5 inches?
Solution
We have the following lengths:
- Base = 12 in
- Height = 5 in
Using the formula for the perimeter, we have:
$latex p=2(12+5)$
$latex p=2(17)$
$latex p=34$
Therefore, the perimeter of the rectangle is equal to 34 in.
EXAMPLE 2
Find the area of a rectangle that has a base of 20 feet and a height of 12 feet.
Solution
We have the following lengths:
- Base = 20 ft
- Height = 12 ft
To calculate the area, we have to multiply the length of the base by the length of the height:
$latex A=20\times 12=240$
Therefore, the area is equal to 240 ft².
EXAMPLE 3
If a rectangle has a base of 15 yards and a height of 8 yards, what is its perimeter?
Solution
We have the following:
- Base = 15 yd
- Height = 8 yd
Applying the formula for the perimeter, we have:
$latex p=2(15+8)$
$latex p=2(23)$
$latex p=46$
The perimeter of the rectangle is equal to 46 yd.
EXAMPLE 4
A rectangular blackboard has the dimensions 140 inches at the base and 90 inches at the height. What is its area?
Solution
We have the following lengths:
- Base = 140 in
- Height = 90 in
Using these lengths in the formula for the area, we have:
$latex A=140\times 90=12600$
The area of the blackboard is 12,600 in².
EXAMPLE 5
A rectangle has a perimeter of 54 inches and a base of 10 inches. Determine its height.
Solution
We have the following:
- Perimeter = 54 in
- Base = 10 in
In this case, we have the perimeter, and we want to determine the length from the height. Therefore, we use the perimeter formula and solve for a:
$latex p=2(b+a)$
$latex 54=2(10+a)$
$latex 54=20+2a$
$latex 2a=34$
$latex a=17$
The length of the height of the rectangle is equal to 17 in.
EXAMPLE 6
A rectangle has an area equal to 120 in². If its base is 20 inches long, what is the length of its height?
Solution
We have the following information:
- Base = 20 in
- Area = 120 in²
We can use the formula for the area and solve for the height:
height $latex =\frac{120}{20}=6$
Therefore, the height is 6 inches long.
EXAMPLE 7
If a rectangle has a base of 18 yards and a height of 12 yards, what is its perimeter?
Solution
We have the following lengths:
- Base = 18 yd
- Height = 12 yd
Using the formula for the perimeter, we have:
$latex p=2(18+12)$
$latex p=2(30)$
$latex p=60$
The perimeter of the rectangle is equal to 60 yards.
EXAMPLE 8
Find the area of a square that has sides with a length of 8 ft.
Solution
A square is a special case of a rectangle, in which its height is equal to its base. Therefore, we have:
- Base = 8 m
- Height = 8 m
Using the formula for the area of a rectangle, we have:
$latex A=8\times 8=64$
Therefore, the area of the square is 64 ft².
EXAMPLE 9
What is the length of the base of a rectangle that has a height of 8 feet and a perimeter of 46 feet?
Solution
We have the following:
- Height = 8 ft
- Perimeter = 46 ft
We use the formula for the perimeter and solve for the base:
$latex p=2(b+a)$
$latex 46=2(b+8)$
$latex 46=2b+16$
$latex 2b=30$
$latex b=15$
The base of the rectangle has a length of 15 ft.
EXAMPLE 10
The area of a square is equal to 6400 in². What is the length of one of its sides?
Solution
We have the following:
- Area = 6400 in²
In a square, the base and height have the same length, so we can use the formula A = side². Thus, we substitute to get:
lado $latex =\sqrt{8100}=80$
Therefore, the length of one of the sides is equal to 80 inches.
Perimeter and area of a rectangle - Practice problems
See also
Interested in learning more about the perimeter and area of geometric figures? Take a look at these pages:
