An equilateral triangle is a triangle in which all sides are equal and all angles are \( 60^\circ \). The area of an equilateral triangle is the part of the plane occupied by the figure. It is measured in square units. For example, these can be \( \text{mm}^2 \), \( \text{cm}^2 \), \( \text{m}^2 \), and other units of area.
In this article, we will look at how to find the area of an equilateral triangle, which formula is used to calculate it, and how to apply it in practice. In addition, we will go through a few examples to help you understand the topic better.
Area of an Equilateral Triangle: Main Formula
How do you find the area of an equilateral triangle? For any triangle, the area is calculated as the product of the base and the height divided by \( 2 \). However, for an equilateral triangle, there is a convenient direct formula:
\[
A=\frac{\sqrt{3}\cdot AB^2}{4},
\]
where \( A \) is the area of the equilateral triangle.
So, to find the area of an equilateral triangle, you need to square its side, multiply it by the square root of three, and divide by \( 4 \).
Proof of the Formula
Where does this formula come from? It can be obtained from the general formula for the area of a triangle. To do this, we need the side length and the height of the equilateral triangle.
Recall that the height of an equilateral triangle is calculated by the formula:
\[
BH=\frac{\sqrt{3}\cdot AB}{2}.
\]
Now let us substitute the value of \( BH \) into the general formula for the area of a triangle:
\[
A=\frac{AB\cdot BH}{2}=\frac{AB\cdot \frac{\sqrt{3}\cdot AB}{2}}{2}=\frac{\sqrt{3}\cdot AB^2}{4}.
\]
So, we obtain the formula for the area of an equilateral triangle.
Note. If we denote the side of an equilateral triangle by the letter \( a \), then the formula can be written more conveniently as:
\[
A=\frac{\sqrt{3}\cdot a^2}{4}.
\]
Area of an Equilateral Triangle: Examples with Answers
To make the formula clearer, let us look at a few examples. Try to solve them on your own first, and then check the answer.
Example 1. Find the area of an equilateral triangle if its side is \( 4 \) cm
We know that the side of the triangle is \( 4 \) cm. Let us substitute this value into the formula:
\[
A=\frac{\sqrt{3}\cdot a^2}{4}=\frac{\sqrt{3}\cdot 4^2}{4}\approx 6.928.
\]
So, the area of the equilateral triangle is \( 6.928\ \text{cm}^2 \).
Example 2. Find the area of an equilateral triangle with side \( 10 \) cm
Here the side of the triangle is \( 10 \) cm. Let us substitute this value into the area formula:
\[
A=\frac{\sqrt{3}\cdot a^2}{4}=\frac{\sqrt{3}\cdot 10^2}{4}\approx 43.301.
\]
So, the area of the equilateral triangle is \( 43.301\ \text{cm}^2 \).
Example 3. The area of an equilateral triangle is \( 56\ \text{cm}^2 \). Find the side of the triangle
In this example, the area is known, and we need to find the side of the triangle. Let us use the formula:
\[
A=\frac{\sqrt{3}\cdot a^2}{4}.
\]
Now substitute the known value:
\[
A=\frac{\sqrt{3}\cdot a^2}{4},\qquad 56=\frac{\sqrt{3}\cdot a^2}{4},\qquad a\approx 11.372.
\]
So, the side length of the equilateral triangle is \( 11.372 \) cm.
Example 4. Find the area of an equilateral triangle if its perimeter is \( 63 \) cm
In this case, we first need to find the side of the triangle. Recall that the perimeter of an equilateral triangle is calculated by the formula:
\[
P=3\cdot a.
\]
Substitute the known perimeter value:
\[
P=3\cdot a,\qquad 63=3\cdot a,\qquad a=21.
\]
Now that the side is known, let us find the area:
\[
A=\frac{\sqrt{3}\cdot a^2}{4}=\frac{\sqrt{3}\cdot 21^2}{4}\approx 190.959.
\]
So, the area of the equilateral triangle is \( 190.959\ \text{cm}^2 \).
What to Read Next: Helpful Topics to Continue With
After learning about area, it is also worth looking at other materials about the equilateral triangle. This will help you better understand its properties and see how the main formulas are connected.
- Equilateral Triangle: Definition and Properties — Learn about the main properties of this figure and how it differs from other triangles.
- Height of an Equilateral Triangle: Formulas and Examples — Understand how to find the height of an equilateral triangle and where this formula is used in problems.
- Perimeter of an Equilateral Triangle: Formulas and Examples — Learn how to calculate the perimeter of an equilateral triangle and apply this formula in practice.
Area of an Equilateral Triangle: From Geometry to Programming
This flowchart clearly shows how the topic of “area of an equilateral triangle” is connected not only with geometry but also with programming. If you are interested not only in substituting values into a formula but also in seeing how a mathematical idea turns into a working program, try implementing this algorithm in your favorite programming language. This approach helps you understand the formula better, see how area calculations are automated when the side is known, and at the same time practice working with input and output data. Even such a short program shows very clearly how mathematics and programming complement each other in practice.
