How to calculate the area of an equilateral triangle

Instructor: Yuanxin (Amy) Yang Alcocer Show bio

Table of Contents Show

• Steps to Solve
• Finding the Area
• Another Example
• Register to view this lesson
• Unlock Your Education
• Resources created by teachers for teachers
• Calculating the Area
• Deriving the Area Expression
• Using the Area of a Triangle
• Using Trigonometry
• What is a formula of equilateral triangle?
• How do you find the area of a triangle with 3 equal sides?

Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.

An equilateral triangle is a triangle that has three sides of the same length. Explore the steps in finding the area of an equilateral triangle and learn to calculate the area through solved examples. Updated: 01/05/2022

Steps to Solve

To find the area of an equilateral triangle, or a triangle with 3 equal sides, all you have to do is to follow these steps.

Step 1

Use this formula for the area of an equilateral triangle:

Area Formula

In this case, the upper case A is the area and the lower case a is the length of the sides, which again, are all equal, hence the same letter being used.

Step 2

Plug in your value for the length of a side of your equilateral triangle and evaluate.

That is it! With a simple formula like this, calculating the area is really easy. Just plug in to the formula and evaluate!

Let's see this in action.

• Video
• Quiz
• Course

Finding the Area

Find the area of this equilateral triangle.

Problem

Step 1

The first step is to remember the formula, which is the square root of 3 divided by 4 multiplied by the square of the side of the equilateral triangle.

Area Formula

Step 2

The next step is to plug in the value for the side of the equilateral triangle. For this triangle, the side is 3 inches long. So, you plug in 3 for the variable s. Square that and you get 9. Multiplying the 9 by the square root of 3 and dividing by 4, you get 3.9 inches squared. And you are done!

Another Example

Let's try another one.

Find the area of an equilateral triangle with sides that measure 6 centimeters.

Step 1

First, write down the formula. The square root of 3 divided by 4 multiplied by the square of the side of the equilateral triangle.

Step 2

Your side measures 6 centimeters, so you plug in 6 to the formula. 6 squared is 36. Multiplying that by the square root of 3 and dividing by 4, you get 15.6 centimeters squared. And you are done.

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An equilateral triangle

Calculating the Area

Recall that the regular formula for the area of a triangle is

{eq}\hspace{2em} A = \frac{1}{2}bh {/eq},

where {eq}b {/eq} is the base of the triangle and {eq}h {/eq} is its height. This formula can be used to calculate the area of an equilateral triangle since it applies to triangles of all sizes and shapes.

However, there is also a quicker formula that applies "only to equilateral triangles. The formula for the area of an equilateral triangle is

{eq}\hspace{2em} A = \frac{a^2\sqrt3}{4} {/eq},

where {eq}a {/eq} is the length of the triangle's sides.

An equilateral triangle with sides of length 4

For example, to calculate the area of the equilateral triangle above, plug in {eq}a=4 {/eq} into the area formula and simplify the result.

{eq}\hspace{2em} A = \frac{4^2\sqrt3}{4} = \frac{16\sqrt3}{4} = 4\sqrt3 {/eq}

Deriving the Area Expression

This formula can be derived either by using the area formula from above, {eq}A = \frac{1}{2}bh {/eq}, or by using the trigonometric identity {eq}A = \frac{1}{2}ab \sin C {/eq}, which uses the side lengths of two sides and the measure of the angle between them.

Using the Area of a Triangle

For the first method, start by drawing a median down the middle of the triangle, dividing it into two congruent right triangles.

Equilateral triangle with median

If the sides of the equilateral triangle are all of length {eq}a {/eq}, then each half of the base are of length {eq}\frac{1}{2}a {/eq}. Moreover, you can find the three angles of the right triangles as follows: they are right triangles ({eq}90^\circ {/eq}) and the angles on the left and right are each {eq}60^\circ {/eq}, so the remaining angle must be {eq}30^\circ {/eq}. In other words, the right triangles are 30-60-90 triangles, which means that their three sides are in the ratio {eq}1 {:} \sqrt3 {:} 2 {/eq}. By this ratio, the median must be {eq}\sqrt3 {/eq} times the bottom side ({eq}\frac{1}{2}a {/eq}), which gives you {eq}\frac{1}{2}a\sqrt3 {/eq}, as shown in the figure above. (This length can also be calculated by finding {eq}a\sin60^\circ {/eq} or {eq}a\cos30^\circ {/eq}.)

Finally, to find the area of the equilateral triangle, just apply the normal area formula with {eq}a {/eq} for the base and {eq}\frac{1}{2}a\sqrt3 {/eq} for the height.

{eq}\hspace{2em} A = \frac{1}{2}bh = \frac{1}{2}(a)(\frac{1}{2}a\sqrt3) = \frac{a^2\sqrt3}{4} {/eq}

Using Trigonometry

Alternatively, you can derive the formula for the area of an equilateral triangle from the trigonometric equation {eq}A = \frac{1}{2}ab \sin C {/eq}. Begin by labeling the sides and angles of an equilateral triangle.

An equilateral triangle with labeled angles and sides

Because this is an equilateral triangle, you know that {eq}C = 60^\circ {/eq} and the side lengths {eq}a {/eq} and {eq}b {/eq} are equal. Plug in {eq}a {/eq} for both sides in the trigonometric equation as well as {eq}60^\circ {/eq} for {eq}C {/eq}.

What is a formula of equilateral triangle?

How do you find the area of a triangle with 3 equal sides?