Equilateral Triangle Calculator

An equilateral triangle has three equal sides and three 60 degree angles, so a single length fixes every other measurement. The height is not the side. It is about 87 percent of it, because the peak sits above the middle of the base.

An inscribed circle touches all three sides; its radius is s√3/6. A circumcircle passes through the three vertices and has radius s√3/3, twice the inradius. The height is three times the inradius.

Consequently, the largest circular opening that fits inside the triangle has a diameter equal to two thirds of its height. Using the full height as an opening diameter would cross the sloping sides. For the existing side-six example, the two circle radii are √3 and 2√3.

The incircle occupies π/(3√3), about 60.46%, of the triangle’s area. In the other direction, the triangle occupies 3√3/(4π), about 41.35%, of its circumcircle’s area. Those fractions follow from the stated radius and area equations and are independent of the side length.

They distinguish a circle cut from a triangular blank from a triangle cut from a circular blank. Area changes with the square of side length, while height, perimeter and both radii change in direct proportion. An assumed 5% side increase multiplies area by 1.05² = 1.1025, a 10.25% increase.

A 5% side decrease multiplies it by 0.95² = 0.9025, a 9.75% decrease. Equal positive and negative length changes therefore do not produce equal area changes.

Verify that all three sides match. If two match and the third differs, the triangle is isosceles and the height comes out different.

See the triangle area, right triangle and law of sines calculators for other triangles, the regular polygon page for the family of equal-sided shapes, and the triangular prism page for solids. The Pythagorean theorem calculator gives the height, and the circle area page gives the inscribed circle's area.