The area of an equilateral triangle is the amount of space that the triangle occupies in two-dimensional space. Recall that an equilateral triangle is a triangle that has three equal sides and in which all its internal angles measure 60°. We can calculate the area of this type of triangle simply by using the length of one of its sides.
Here, we will earn about the formula that we can use to calculate the area of an equilateral triangle. Also, we will look at some solved examples in which we will apply this formula to find the answer.
GEOMETRY
Relevant for…
Learning about the area of an equilateral triangle with examples.
GEOMETRY
Relevant for…
Learning about the area of an equilateral triangle with examples.
Formula for the area of an equilateral triangle
The formula to calculate the area of an equilateral triangle is given by:
| $latex A= \frac{ \sqrt{3}}{4}{{a}^2}$ |
where a represents the length of one of the sides of the equilateral triangle.
Derivation of this formula
We can take an equilateral triangle with sides of length a. Then, we draw a bisector perpendicular to the base with height h:
Now, we have the following formula to calculate the area of any triangle:
$latex \text{Area}= \frac{1}{2} \times \text{base} \times \text{height}$
Here, the base is equal to “a” and the height is equal to “h“.
If we apply the Pythagorean theorem on the triangle, we have:
$latex {{a}^2}={{h}^2}+{{( \frac{a}{2})}^2}$
⇒ $latex {{h}^2}={{a}^2}- \frac{{{a}^2}}{4}$
⇒ $latex {{h}^2}=\frac{3{{a}^2}}{4}$
⇒ $latex h=\frac{\sqrt{3}~a}{2}$
We have found the value of the height, h. Using this expression in the area formula, we have:
$latex \text{Area}= \frac{1}{2} \times \text{base} \times \text{height}$
$latex A=\frac{1}{2}\times a \times \frac{\sqrt{3}~a}{2}$
⇒ $latex A=\frac{\sqrt{3}~{{a}^2}}{4}$
Area of an equilateral triangle – Examples with answers
Apply the formula for the area of an equilateral triangle above to solve the following examples. Each example has its respective solution, but it is recommended that you try to solve the exercises yourself before looking at the solution.
EXAMPLE 1
Find the area of an equilateral triangle that has sides with a length of 10 m.
Solution
We can use the formula for the area with the length of 10 m:
$latex A= \frac{ \sqrt{3}}{4}{{a}^2}$
$latex A= \frac{ \sqrt{3}}{4}({{10}^2})$
$latex A= \frac{ \sqrt{3}}{4}(100)$
$latex A=43.3$
The area of the equilateral triangle is 43.3 m².
EXAMPLE 2
What is the area of an equilateral triangle that has sides with a length of 14 m?
Solution
We have a length of 14 m. Therefore, using the formula with this value, 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 84.87 m².
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EXAMPLE 3
An equilateral triangle has sides of length 15 m. What is its area?
Solution
We substitute $latex a=15$ 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 97.43 m².
EXAMPLE 4
An equilateral triangle has an area of 35.07 m². What is the length of its sides?
Solution
In this case, we start from the area and want to find the length of the sides. Therefore, we 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 length of the sides of the triangle is 9 m.
EXAMPLE 5
What is the length of the sides of an equilateral triangle that has an area of 73.18 m²?
Solution
We use the area formula and solve for the length:
$latex A= \frac{ \sqrt{3}}{4}{{a}^2}$
$latex 73.18= \frac{ \sqrt{3}}{4}{{a}^2}$
$latex 73.18=0.433{{a}^2}$
$latex {{a}^2}=169$
$latex a=13$
The length of the sides of the triangle is 13 m.
Area of an equilateral triangle – Practice problems
Put into practice the use of the learned formula to solve the following problems. If you need help with this, you can look at the solved examples above.
See also
Interested in learning more about equilateral triangles? Take a look at these pages:
