Area Of Equilateral Triangle Calculator

What this calculator does

An equilateral triangle is the perfectly even one: all three sides the same length, and all three angles 60 degrees. Because everything about it is equal, a single number, the length of one side, settles the whole shape.

So that is all this tool asks for. Type the side, pick a unit, and it gives you the area.

Using the calculator

1. Type the side length.
2. Pick its unit.
3. Press Calculate.

The side has to be a positive number. There is only the one input, since all three sides of an equilateral triangle are the same.

The formula | area = (√3 ÷ 4) × side²

The area of an equilateral triangle is:

area = (√3 ÷ 4) × side²

Square the side, then multiply by √3 ÷ 4, which works out to about 0.433. That gives a quick mental shortcut: square the side and take a little under half of it.

Where the √3 comes from

The √3 is not pulled out of thin air. Drop a straight line from the top corner down to the middle of the base, and it splits the equilateral triangle into two matching right triangles. Working the height out with the Pythagorean theorem gives:

height = (√3 ÷ 2) × side

Now feed that into the ordinary triangle formula, ½ × base × height: ½ × side × (√3 ÷ 2 × side) = (√3 ÷ 4) × side². That is the whole story of the formula, and the 60 degree corners are what bring the √3 in.

Units and rounding

The area comes out in square units of the side's unit, since it is built from a length times a length. A tidy result is shown in full, and a long decimal is rounded to three places.

A worked example | a 6 cm side

Say the side is 6 cm.

1. Square it: 6² = 36.
2. Multiply by √3 ÷ 4 (about 0.433): 36 × 0.4330 = 15.588.

So the area is about 15.588 cm². If you are curious, the height that goes with it is (√3 ÷ 2) × 6 = 5.196 cm.

Questions people ask

What is the area of an equilateral triangle with a 6 cm side?

About 15.588 cm². Square the side and multiply by √3 ÷ 4: (√3 ÷ 4) × 36.

Why is there a √3 in the formula?

Because the height of an equilateral triangle is (√3 ÷ 2) × side, which comes from splitting it down the middle and using the Pythagorean theorem. Put that height into ½ × base × height and the √3 ÷ 4 falls out.

Do I need to know the height?

No. The side alone is enough, since the height of an equilateral triangle is fixed by its side. The formula works the height out for you.

Is this the same as Heron's formula with three equal sides?

Yes. Put three equal sides into Heron's formula and it simplifies to (√3 ÷ 4) × side². This tool just goes straight there from the single side.

Why is the answer in square units?

Because area is a length times a length. A side in centimetres gives an area in square centimetres.

References

A note on where this comes from. The equilateral triangle is the very first shape Euclid builds in the Elements, and its area follows from its height, which the Pythagorean theorem fixes at (√3 ÷ 2) times the side. That √3 is an irrational number, so the area of an equilateral triangle with a whole-number side is never a tidy round figure. For further reading, see Equilateral triangle.

1. Euclid, Elements, Book I, Proposition 1 (c. 300 BCE), the construction of an equilateral triangle on a given line.
2. The Pythagorean theorem, used to find the height (√3 ÷ 2) × side from which the area formula follows.