# Equilateral Triangle Calculator

By Hanna Pamuła, PhD candidate
Last updated: Oct 19, 2020

The equilateral triangle calculator will help you with calculations of the regular triangle parameters. Whether you are looking for the equilateral triangle area, its height, perimeter, circumradius or inradius, this great tool is a safe bet. Scroll down to read more about useful formulas and to get to know what is an equilateral triangle.

## What is an equilateral triangle?

The equilateral triangle, also called a regular triangle, is a triangle with all three sides equal. What are the other important properties of that specific regular shape? • all three internal angles are congruent to each other and all of them are equal to 60°;
• the altitudes, the angle bisectors, the perpendicular bisectors and the medians coincide.

The equilateral triangle is a special case of an isosceles triangle having not just two, but all three sides equal.

## Equilateral triangle area and height

The formula for a regular triangle area is equal to squared side times square root of 3 divided by 4:

`area = (a² * √3)/ 4`

and the equation for the height of equilateral triangle look as follows:

`h = a * √3 / 2`, where `a` is a side of the triangle.

But do you know where the formulas come from? You can find them in at least two ways: deriving from Pythagorean theorem or using trigonometry.

1. Using Pythagorean theorem

• The basic formula for triangle area is side `a` (base) times the height `h`, divided by 2:

`area = (a * h) / 2`

• Height of the equilateral triangle is splitting the equilateral triangle into two right triangles. One leg of that right triangle is equal to height, other leg is half of the side, and the hypotenuse is the equilateral triangle side.

`(a/2)² + h² = a²`

After simple transformations we get a formula for the height of the equilateral triangle:

`h = a * √3 / 2`

• Substituting `h` into the first area formula, we obtain the equation for the equilateral triangle area:

`area = a² * √3 / 4`

2. Using trigonometry

• Let's start from the trigonometric triangle area formula:

`area = (1/2) * a * b * sin(γ)` , where `γ` is the angle between sides

• We remember that all sides and all angles are equal in the equilateral triangle, so the formula simplifies to: `area = 0.5 * a * a * sin(60°)`

• What is more, we know that the sine of 60° is √3/2, so the formula for equilateral triangle area is:

`area = (1/2) * a² * (√3 / 2) = a² * √3 / 4`

Height of the equilateral comes from sine definition:

`h / a = sin(60°)` so `h = a * sin(60°) = a * √3 / 2`

## Equilateral triangle perimeter, circumcircle and incircle radius

You can easily find the perimeter of an equilateral triangle by adding all triangles sides together. The regular triangle has all sides equal, so the formula for the perimeter is:

`perimeter = 3 * a`

How to find the radius of the circle circumscribing the three vertices and the inscribed circle radius?

`circumcircle_radius = 2 * h / 3 = a * √3 / 3`

`incircle_radius = h / 3 = a * √3 / 6`

## How can I use the equilateral triangle calculator?

Let's take the example from everyday life: we want to find all the parameters of the yield sign.

1. Type the given value into right box. Assume we have the sign with 36 in side length.
2. The equilateral triangle calculator finds the other values in no time. Now we know that:
• yield sign height is 31.2 in
• its area equals 561 in²
• perimeter: 108 in
• circumcircle radius is 20.8 in
1. Check out our tool flexibility. Refresh the calculator, and type in the other parameter, e.g. perimeter. It's working this way as well, isn't that cool?
Hanna Pamuła, PhD candidate Side (a)
in
Height (h)
in
Area
in²
Perimeter
in
in
in
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