When you are asked to find the area of an equilateral triangle, most problems will not give you the height of the triangle. This is because equilateral triangles have some unique properties that make it possible to calculate the area from just the side length.

You can use any of these methods to find the area of an equilateral triangle:

- Use the Pythagorean Theorem to find the height of the triangle and then calculate the area.
- Use the Law of Sines to find the height of the triangle and then calculate the area.
- Use Heron’s Formula to calculate the area.

All three methods will give you the same formula for the area of an equilateral triangle.

## Area of an Equilateral Triangle Formula

In this formula, the A represents the area and the x represents the side length of the equilateral triangle.

\[A=\frac{\sqrt{3}}{4}x^2\]

## How to Find the Area of an Equilateral Triangle

- Identify the side length of the equilateral triangle.
- Evaluate the formula \(A=\frac{\sqrt{3}}{4}x^2\).
- Make sure the answer is labeled with the correct units.

## Examples

*What is the side length of the triangle?*

The side length of this equilateral triangle is 5 in.

*What is the evaluated area formula? *

\[A=\frac{\sqrt{3}}{4}(5 \, in)^2\]

Five squared equals twenty-five.

\[A=\frac{\sqrt{3}}{4}25 \, in^2\]

The square root of three divided by four is approximately equal to 0.43301270189

\[A=(0.43301270189)25 \, in^2\]

Twenty-five times 0.43301270189 is equal to 10.8253175473.

\[A=10.8253175473 \, in^2\]

*What are the correct units for the area of this triangle?*

The side lengths of the triangle were given in inches, so the area of the triangle should be measured in square inches.

The area of this triangle is \(10.8253175473 \, in^2\)

*What is the side length of the triangle?*

The side length of this equilateral triangle is 12 cm.

*What is the evaluated area formula? *

\[A=\frac{\sqrt{3}}{4}(12 \, cm)^2\]

Twelve squared equals one hundred forty-four.

\[A=\frac{\sqrt{3}}{4}144 \, cm^2\]

The square root of three divided by four is approximately equal to 0.43301270189

\[A=(0.43301270189)144 \, cm^2\]

One hundred forty-four times 0.43301270189 is equal to 62.3538290725.

\[A=62.3538290725 \, cm^2\]

*What are the correct units for the area of this triangle?*

The side lengths of the triangle were given in centimeters, so the area of the triangle should be measured in square centimeters.

The area of this triangle is \(62.3538290725 \, cm^2\).

## Why It Works

The Area of an Equilateral Triangle Formula is a special case of the Area of a Triangle Formula.

The area of any triangle is \(\frac{1}{2}bh\) where b is the base of the triangle and h is the height of the triangle that is perpendicular to the base.

To find the area of an equilateral triangle, you just have to identify the base and height of the equilateral triangle and plug those values into the Area of a Triangle Formula.

### Area of a Triangle Formula

\[A=\frac{1}{2}bh\]

The variable x represents the length of one side of the equilateral triangle.

All the sides of an equilateral triangle are the same length and any of these sides can be used as the base of the triangle.

\[{\yellow Base}={\yellow x}\]

The height of an equilateral triangle always equals \(\frac{\sqrt{3}}{2}x\) because of the Pythagorean Theorem and the Law of Sines.

\[{\red Height}={\red \frac{\sqrt{3}}{2}x}\]

The area of any triangle is \({\green \frac{1}{2}}{\yellow b}{\red h}\). For an equilateral triangle…

\[{\yellow Base}={\yellow x}\]

\[{\red Height}={\red \frac{\sqrt{3}}{2}x}\]

So, the area of an equilateral triangle is…

\[A={\green \frac{1}{2}}({\yellow x})({\red \frac{\sqrt{3}}{2}x})\]

When you multiply the fractions and variables together, the simplified formula for the area of an equilateral triangle is…

\[A=\frac{\sqrt 3}{4}x^2\]

Alternatively, you could also prove this formula by plugging the side lengths into Heron’s Formula.

### Heron's Formula

\[A=\sqrt{s(s-a)(s-b)(s-c)}\]

The perimeter of an equilateral triangle is…

\[{\red x}+{\blue x}+{\green x}=3x\]

The semiperimeter is half of that.

\[s=\frac{3x}{2}\]

When s is the semiperimeter and a, b, and c are the side lengths of the triangle, Heron’s formula says the area of a triangle is…

\[\sqrt{{\yellow s}({\yellow s}-{\red a})({\yellow s}- {\blue b})({\yellow s}-{\green c})}\]

For equilateral triangles, \(\yellow s=\frac{3x}{2}\), \(\red a=x\), \(\blue b=x\), and \(\green c=x\).

So, Heron’s formula can be evaluated as…

\[\sqrt{{\yellow \frac{3x}{2}}({\yellow \frac{3x}{2}}-{\red x})({\yellow \frac{3x}{2}}- {\blue x})({\yellow \frac{3x}{2}}-{\green x})}\]

We can use fraction subtraction to simplify inside all the parentheses which gives us…

\[\sqrt{{\yellow \frac{3x}{2}}(\frac{x}{2})(\frac{x}{2})(\frac{x}{2})}\]

Then when we multiply all the fractions together, we get…

\[\sqrt{\frac{3x^4}{16}}\]

The square root of 16 is 4. The square root of \(x^4\) is \(x^2\) and the square root of 3 cannot be simplified.

So, the simplified formula for the area of an equilateral triangle is…

\[\frac{\sqrt{3}}{4}x^2\]