Equilateral Triangle Calculator

Created by Hanna Pamuła, PhD
Reviewed by Bogna Szyk and Adena Benn
Last updated: Oct 07, 2022

The equilateral triangle calculator will help you with calculations of standard 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 valuable formulas (such as to calculate the height of an equilateral triangle) and learn what an equilateral triangle is.

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?

Image of equilateral traingle, with sides, angles, and height marked.
  • 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. If you would like to learn more about the isosceles triangle, our isosceles triangle calculator is just the tool you need.

Equilateral triangle area and height

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

area = (a² × √3)/ 4

and the equation for the height of an equilateral triangle looks 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 the Pythagorean theorem (discussed in our Pythagorean theorem calculator) 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 derived by splitting the equilateral triangle into two right triangles. See our right triangle calculator to learn more about right triangles.

One leg of that right triangle is equal to height, another 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 with the trigonometric triangle area formula:

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

    The height of the equilateral comes from the 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. This 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 an example from everyday life: we want to find all the parameters of the yield sign.

  1. Type the given value into the correct box. Assume we have a sign with a 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; and
    • Incircle radius 10.4 in.
  3. 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?

FAQ

How do I find the area of an equilateral triangle?

To find the area of an equilateral triangle, follow the given instructions:

  1. Take the square root of 3 and divide it by 4.

  2. Multiply the square of the side with the result from step 1.

  3. Congratulations! You have calculated the area of an equilateral triangle.

How do I find the height of an equilateral triangle?

To find the height of an equilateral triangle, proceed as follows:

  1. Take the square root of 3 and divide it by 2.

  2. Multiply the result from step 1 with the length of the side.

  3. You will get the height of the equilateral triangle.

How do I find the perimeter of an equilateral triangle with a side length of 8 cm?

The perimeter of the given triangle is 24 cm.

To calculate the perimeter of an equilateral triangle, we need to multiply its side length by 3. The length of each side of the given triangle is 8 cm. Hence its perimeter will be 3 × 8 cm = 24 cm.

Can a right triangle be equilateral?

No, a right triangle can't be an equilateral triangle. One of the angles in a right triangle is 90°. Since the sum of all the interior angles in a triangle is 180°, the other two angles in a right triangle are always less than 90°.

According to the definition of equilateral triangles, all the internal angles are equal. Hence, a right triangle can never be equilateral triangle.

Hanna Pamuła, PhD
Image of equilateral triangle, with sides, angles, and height marked.
Side (a)
in
Height (h)
in
Area
in²
Perimeter
in
Circumcircle radius
in
Incircle radius
in
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