The perimeter of an equilateral triangle is the length of the triangle's outline. On the other hand, the area is a measure of the space occupied by the triangle. We can find the perimeter of an equilateral triangle by adding the lengths of its three sides, and we can find its area by multiplying the length of its base by its height and dividing by 2.
In this article, we will learn all about the perimeter and area of an equilateral triangle. We will explore its formulas and apply them to solve some practice problems.
GEOMETRY
Relevant for...
Learning about the perimeter and area of an equilateral triangle.
GEOMETRY
Relevant for...
Learning about the perimeter and area of an equilateral triangle.
How to find the perimeter of an equilateral triangle?
To calculate the perimeter of an equilateral triangle, we have to add the lengths of its three sides. Recalling that an equilateral triangle has all its sides of the same length, we only have to multiply the length of one of the sides by 3:
| $latex p=3a$ |
where, a is the length of one side of the triangle.
This means that we only need to know the length of one of the sides of the equilateral triangle to calculate its perimeter.
How to find the area of an equilateral triangle?
To calculate the area of any triangle, we can multiply its base by its height and divide by 2. In the case of equilateral triangles, we can use the following formula to calculate their area:
| $latex A= \frac{ \sqrt{3}}{4}{{a}^2}$ |
where, a is the length of one of the sides of the equilateral triangle.
Proof of the formula for the area of an equilateral triangle
To prove the formula for the area of an equilateral triangle, we are going to use the following diagram, where we draw a bisector perpendicular to the base with height h:
We recall that the area of any triangle can be calculated with the following formula:
$latex \text{Area}= \frac{1}{2} \times \text{base} \times \text{height}$
Here, the base is equal to "a" and the height is equal to "h".
Using the Pythagorean theorem to calculate the height, we have:
$latex {{a}^2}={{h}^2}+{{( \frac{a}{2})}^2}$
⇒ $latex {{h}^2}={{a}^2}- \frac{{{a}^2}}{4}$
⇒ $latex {{h}^2}=\frac{3{{a}^2}}{4}$
⇒ $latex h=\frac{\sqrt{3}~a}{2}$
Now that we have an expression for h, we can use it in the formula for the area of a triangle:
$latex \text{Area}= \frac{1}{2} \times \text{base} \times \text{height}$
$latex A=\frac{1}{2}\times a \times \frac{\sqrt{3}~a}{2}$
⇒ $latex A=\frac{\sqrt{3}~{{a}^2}}{4}$
Area and perimeter of an equilateral triangle - Examples with answers
EXAMPLE 1
Find the perimeter of an equilateral triangle that has sides with a length of 5 inches.
Solution
We use the formula for the perimeter with the value $latex a=5$. Therefore, we have:
$latex p=3a$
$latex p=3(5)$
$latex p=15$
The perimeter of the equilateral triangle is equal to 15 inches.
EXAMPLE 2
What is the area of an equilateral triangle that has sides with a length of 10 feet?
Solution
We use the formula for the area with the length a=10:
$latex A= \frac{ \sqrt{3}}{4}{{a}^2}$
$latex A= \frac{ \sqrt{3}}{4}({{10}^2})$
$latex A= \frac{ \sqrt{3}}{4}(100)$
$latex A=43.3$
The area of the equilateral triangle is equal to 43.3 ft².
EXAMPLE 3
Find the perimeter of an equilateral triangle that has sides with a length of 8 yards.
Solution
Using the value $latex a=8$ in the formula for the perimeter, we have:
$latex p=3a$
$latex p=3(8)$
$latex p=24$
The perimeter of the equilateral triangle is equal to 24 yd.
EXAMPLE 4
Find the area of an equilateral triangle that has sides with a length of 14 inches.
Solution
Applying the formula for the area with the given length, we have:
$latex A= \frac{ \sqrt{3}}{4}{{a}^2}$
$latex A= \frac{ \sqrt{3}}{4}({{14}^2})$
$latex A= \frac{ \sqrt{3}}{4}(196)$
$latex A=84.87$
The area of the equilateral triangle is equal to 84.87 in².
EXAMPLE 5
What is the perimeter of an equilateral triangle that has sides with a length of 15 inches?
Solution
Applying the formula for the perimeter with the value $latex a=15$:
$latex p=3a$
$latex p=3(15)$
$latex p=45$
The perimeter of the triangle is equal to 45 inches.
EXAMPLE 6
What is the area of an equilateral triangle that has sides with a length of 15 feet?
Solution
We use the length $latex a=15$ in the formula for the area:
$latex A= \frac{ \sqrt{3}}{4}{{a}^2}$
$latex A= \frac{ \sqrt{3}}{4}({{15}^2})$
$latex A= \frac{ \sqrt{3}}{4}(225)$
$latex A=97.43$
The area of the equilateral triangle is equal to 97.43 ft².
EXAMPLE 7
Find the length of the sides of an equilateral triangle that has a perimeter of 39 ft.
Solution
In this example, we know the perimeter of the triangle, and we have to find the length of one side. Therefore, we use the perimeter formula and solve for a:
$latex p=3a$
$latex 39=3a$
$latex a=13$
The sides of the triangle have a length of 13 ft.
EXAMPLE 8
Find the length of the sides of an equilateral triangle that has an area of 35.07 ft².
Solution
In this case, we know the area, and we need to find the length of the sides. Therefore, we use the formula for the area and solve for a:
$latex A= \frac{ \sqrt{3}}{4}{{a}^2}$
$latex 35.07= \frac{ \sqrt{3}}{4}{{a}^2}$
$latex 35.07=0.433{{a}^2}$
$latex {{a}^2}=81$
$latex a=9$
The sides of the triangle have a length of 9 ft.
EXAMPLE 9
Find the length of the sides of an equilateral triangle that has a perimeter of 102 in.
Solution
We are going to use the formula for the perimeter with the value $latex p=102$ and solve for a:
$latex p=3a$
$latex 102=3a$
$latex a=34$
The length of one side of the triangle is 34 in.
EXAMPLE 10
Find the length of the sides of an equilateral triangle that has an area of 73.18 ft².
Solution
We use the formula for the area with the given value and solve for a:
$latex A= \frac{ \sqrt{3}}{4}{{a}^2}$
$latex 73.18= \frac{ \sqrt{3}}{4}{{a}^2}$
$latex 73.18=0.433{{a}^2}$
$latex {{a}^2}=169$
$latex a=13$
The length of the sides of the triangle is 13 ft.
Area and perimeter of an equilateral triangle - Practice problems
See also
Interested in learning more about perimeters and areas of geometric figures? Take a look at these pages:
