Equilateral Triangle Calculator

Background.

Three equal sides force three equal angles of 60°, and that leaves an equilateral triangle with a single free number: its size. Know any one measurement — a side, the distance around it, the space it covers, its height, or the radius of a circle drawn inside it or around it — and every other measurement follows. That works in both directions. Someone cutting triangular panels starts from the side and wants the height to set a saw fence; someone filling a triangular corner with 2 m² of paving starts from the area and wants the side, which comes to about 2.1491 m.

All of it traces back to one right triangle. Drop a perpendicular from the top corner and it lands at the midpoint of the base, splitting the shape into two 30-60-90 triangles whose hypotenuse is the side a and whose short leg is a/2. Pythagoras then gives the height as √(a² − a²/4) = (√3/2)a, roughly 0.866 of the side. The area is half the base times that height, (√3/4)a², and the radii of the inscribed and circumscribed circles are one third and two thirds of the height.

Those thirds come from a coincidence no other triangle has. Here the centroid, the incentre, the circumcentre and the orthocentre are a single point, sitting a third of the way up every altitude. So the circumradius is exactly twice the inradius, and the height exactly three times it.

Since √3 is irrational, a triangle with a whole-number side never has a whole-number height or area; only the perimeter stays tidy. The other values are rounded decimals, far finer than any saw or printer can follow, though a hand calculation that used 1.732 for √3 will drift in the fourth decimal place.

What is equilateral triangle calculator?

An equilateral triangle is a triangle whose three sides have the same length. Equal sides sit opposite equal angles, and the angles of any triangle add to 180°, so each angle is 60° and the side length a describes the shape completely. Euclid makes it the first construction in the Elements (Book I, Proposition 1): draw two circles of radius a centred on the ends of a segment of length a, and either point where they cross is the third corner. In this triangle each altitude is also a median, an angle bisector and the perpendicular bisector of the opposite side, so one height answers four geometric questions at once.

How to use this calculator.

1. 01In the first menu, pick the measurement you actually have: side length, perimeter, area, height (altitude), inradius or circumradius.
2. 02Type its value in whatever unit you measured — millimetres, feet, pixels. Lengths come back in that unit and the area in that unit squared, so an area in m² gives a side in metres.
3. 03For a triangular frame around something round, such as a dial or a pipe, choose inradius and enter the circle's radius, which is half its diameter.
4. 04For three points spaced evenly around a circle — bolt holes on a flange, legs under a round tabletop — choose circumradius and enter the radius of that circle; the side output is the straight-line distance between neighbouring points.
5. 05Check the results against the shape's built-in ratios: the circumradius should be exactly twice the inradius and the height exactly three times it.

The formula.

Every option is first turned into the side a. From the perimeter, a = P/3. From the area, a = √(4A/√3). From the height, a = 2h/√3. From the inradius, a = 6r/√3, which is 2√3·r. From the circumradius, a = 3R/√3, which is √3·R. The five outputs are then rebuilt from a: h = (√3/2)a, A = (√3/4)a², P = 3a, r = (√3/6)a and R = (√3/3)a. These are the closed forms in Wolfram MathWorld's entry for the shape; the trigonometric versions, such as h = a sin 60°, give the same values because sin 60° is exactly √3/2. Every length is a fixed multiple of a, so doubling the side doubles them all, but the area grows with a², which is why the area route needs a square root: quadruple the area and the side only doubles. Heron's formula, which knows nothing about equal sides, makes a useful independent check on any area result.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Wolfram Research, MathWorld — ‘Equilateral Triangle’ (live revision retrieved 2026-07-29). Equation (1) h = a sin 60° = ½√3 a; Equation (2) A = ½ah = ¼√3 a²; Equations (3)–(5) r = ½a cot(π/3) = ½a tan(π/6) = ⅙√3 a; Equations (6)–(8) R = ½a csc(π/3) = ½a sec(π/6) = ⅓√3 a. Open access; all four confirmed on retrieval.
2. [2]Wolfram Research, MathWorld — ‘Triangle’ (live revision retrieved 2026-07-29), Equation (18) Δ = ½ah — the general base-and-height area form that the equilateral closed form specialises. Open access.
3. [3]Euclid, Elements, Book I, Proposition 1 — ‘To construct an equilateral triangle on a given finite straight line’, the first proposition in the Elements, carried out with two circles and a straightedge. David E. Joyce web edition of the Heath translation, Clark University; quotation confirmed on retrieval 2026-07-29.
4. [4]Euclid, Elements, Book I, Proposition 47 (the Pythagorean theorem) — used here for the 30-60-90 decomposition that gives the altitude, h = √(a² − (a/2)²) = a√3/2. Joyce web edition, Clark University; retrieved 2026-07-29.