What makes a triangle equilateral?

A triangle is equilateral when all three sides are equal in length, which forces all three interior angles to be exactly 60°. This perfect symmetry means every equilateral triangle is also equiangular and is the most regular polygon possible with three sides.

What is the formula for the area of an equilateral triangle?

Area = (sqrt(3) / 4) * a^2, where a is the side length. Because all sides are equal you only need one measurement. For example, a side of 10 units gives an area of (sqrt(3)/4)*100 = 25*sqrt(3) which is approximately 43.301 square units.

How do I find the height of an equilateral triangle?

The height (altitude from any vertex to the opposite side) is h = a * sqrt(3) / 2. For a side of 10 units the height is 5*sqrt(3) or roughly 8.660 units. This formula comes directly from dropping a perpendicular that splits the base in half and creates a 30-60-90 right triangle.

What is the inradius and how is it different from the circumradius?

The inradius (r) is the radius of the largest circle that fits inside the triangle, touching all three sides. The circumradius (R) is the radius of the circle that passes through all three vertices. For an equilateral triangle R = 2r, so the circumscribed circle is always exactly twice the size of the inscribed one. In formulas: r = a / (2*sqrt(3)) and R = a / sqrt(3).

How can I calculate the side if I only know the area?

Rearrange the area formula: a = sqrt(4A / sqrt(3)). If the area is 100 square units, then a = sqrt(400 / 1.732) which is approximately 15.197 units. The calculator does this rearrangement automatically when you choose "Area" in the Solve-from menu.

Is the equilateral triangle calculator accurate?

Results are computed using IEEE 754 double-precision floating-point arithmetic (the native JavaScript number type) and rounded to eight decimal places for display. For practical engineering or geometry this gives more than enough precision. All calculations run in your browser — no data is transmitted to any server.

Equilateral Triangle Calculator

An equilateral triangle is the most symmetrical of all triangles: every side is equal, every angle is 60°, and every geometric centre (centroid, incentre, circumcentre and orthocentre) coincides at the same point. That perfect regularity means that if you know any one of the six standard measurements — side, height, area, perimeter, inradius or circumradius — you can compute all the others from a single formula chain.

This calculator accepts the value you already have and returns every other property in one step. It is useful for architecture and structural design, classroom geometry, woodworking and carpentry, tiling and tessellation work, and any engineering problem involving equilateral shapes.

How it works

All six properties of an equilateral triangle flow from a single quantity: the side length a. The calculator first converts whichever input you provide back into a, then applies the full set of formulas:

Height: h = a * sqrt(3) / 2
Perimeter: P = 3 * a
Area: A = (sqrt(3) / 4) * a^2
Inradius: r = a / (2 * sqrt(3)) (one-third of the height)
Circumradius: R = a / sqrt(3) (two-thirds of the height)

The reverse derivations used when you provide a non-side input are:

Given height: a = 2h / sqrt(3)
Given area: a = sqrt(4A / sqrt(3))
Given perimeter: a = P / 3

A small SVG diagram above the inputs labels the vertices A, B, C and shows the dashed altitude (height) from A to the midpoint of BC, so the geometry is always visible alongside the numbers.

Worked example

Suppose you are cutting a decorative equilateral triangle with a side length of 20 cm.

Height: 20 * sqrt(3) / 2 = 17.321 cm
Perimeter: 3 * 20 = 60 cm
Area: (sqrt(3) / 4) * 400 = 173.205 cm^2
Inradius: 20 / (2 * sqrt(3)) = 5.774 cm
Circumradius: 20 / sqrt(3) = 11.547 cm

Notice that the circumradius is exactly twice the inradius, which is always true for equilateral triangles. Also note that the centroid lies one-third of the way up from the base, at height h/3 = 5.774 cm — which equals the inradius. These elegant relationships make equilateral triangles appear throughout architecture, Islamic geometric art, and engineering trusses.

Side (a)	Height (h)	Area (A)	Perimeter
5	4.330	10.825	15
10	8.660	43.301	30
20	17.321	173.205	60
50	43.301	1082.531	150

Every figure is calculated entirely in your browser. No measurements are uploaded or stored anywhere.