Equilateral Triangle Calculator

Background.

The Quanta equilateral triangle calculator solves the whole shape from any single measurement you have. Enter the side length, the perimeter, the area, the height, the inradius or the circumradius, and it returns all six. With the default side of 10 it gives a height of 8.6602540378, an area of 43.3012701892, a perimeter of 30, an inradius of 2.8867513459 and a circumradius of 5.7735026919.

An equilateral triangle is the most constrained triangle there is. All three sides are equal and, as a consequence, all three angles are exactly 60°. That leaves precisely one degree of freedom — a scale — which is why one number is enough here when a general triangle needs three. It is also why the inverse questions are as natural as the forward ones: if you know the area of a triangular sail, a paving tile or a plot and want the side to cut it to, this page runs the arithmetic backwards without any algebra on your part.

The construction is the first proposition in Euclid's Elements — Book I, Proposition 1, "To construct an equilateral triangle on a given finite straight line" — solved with two circles and a straightedge before any other result in the book. The measurements follow from a single decomposition: drop the altitude from any vertex and the triangle splits into two congruent 30-60-90 right triangles, each with hypotenuse equal to the side and short leg equal to half of it. The Pythagorean theorem then gives the height as √(a² − a²/4) = a√3 ÷ 2, and every other formula on this page follows from that height and the area it produces.

The results carry three relationships that are worth remembering because they are unique to this shape. The height is exactly three times the inradius. The circumradius is exactly twice the inradius. And the circumradius is two thirds of the height, which is another way of saying that the centre of an equilateral triangle sits two thirds of the way along each median — the centroid, the incentre, the circumcentre and the orthocentre are all the same point, which happens only when a triangle is equilateral. In any other triangle those four centres are distinct, which is why the equilateral case gets its own set of one-line formulas rather than being solved through the general machinery.

One honest caveat belongs beside the numbers. √3 is irrational, so an equilateral triangle can never have a rational side and a rational height at the same time, and never a rational side and a rational area. Cut a tile with a side of exactly 10 centimetres and its height is 8.6602540378… centimetres, an unending decimal. The perimeter is the only derived output that stays whole when the side is whole. Every other figure on this page is exact to the ten decimal places shown and approximate beyond them — which is far finer than any physical tolerance, but worth knowing when you are checking someone else's arithmetic and the last digit does not match.

The shape turns up wherever equal spacing is needed in a plane: triangular truss panels, hex-grid subdivisions in mapping and game boards, close-packed circle arrangements whose centres form equilateral triangles, the geometry of a three-legged stand, and the standard triangular route problems in navigation. In every one of them the question is the same — you know one measurement and need the rest — and that is the whole job of this page.

What is equilateral triangle calculator?

An equilateral triangle is a triangle whose three sides are all the same length. Because equal sides face equal angles, it follows immediately that all three angles are equal, and since the angles of any triangle sum to 180°, each one is exactly 60°. Euclid opens the Elements with its construction: Book I, Proposition 1, "To construct an equilateral triangle on a given finite straight line", drawn with two circles of equal radius meeting at the apex. Wolfram MathWorld's Equilateral Triangle entry collects the closed forms for a triangle of side a: the altitude is h = a sin 60° = ½√3 a (its Equation 1); the area is A = ½ah = ¼√3 a² (Equation 2); the inradius is r = ½a cot(π/3) = ⅙√3 a (Equations 3 to 5); and the circumradius is R = ½a csc(π/3) = ⅓√3 a (Equations 6 to 8). The perimeter is 3a by definition. Two consequences distinguish this shape from every other triangle: the circumradius is exactly twice the inradius, and the height is exactly three times the inradius. Both follow from the fact that the triangle's centroid, incentre, circumcentre and orthocentre coincide at a single point — a coincidence that occurs if and only if the triangle is equilateral. The altitude is simultaneously a median, an angle bisector and a perpendicular bisector of the opposite side, which is also unique to this case and is why the same number, 8.6602540378 for a side of 10, answers four different geometric questions at once.

How to use this calculator.

1. 01Choose which measurement you have from the dropdown. All six routes lead to the same triangle, because one number fixes the whole shape.
2. 02Enter the value in any unit. Lengths come back in that unit, the area in that unit squared, and no conversion happens anywhere.
3. 03Read the side length at the top — that is the answer to the most common inverse question, "my triangle has this area, how long is each side?".
4. 04Read the height if you need a perpendicular depth for a cut, a fill, or a material estimate. In this shape it is also the median and the angle bisector.
5. 05Read the inradius and circumradius if you are fitting a circle inside the triangle or drawing one through its corners — for example seating a round object in a triangular frame, or laying out three points on a circle.
6. 06Check the identities if you want to verify the arithmetic by hand: the circumradius should come out at exactly twice the inradius, and the height at exactly three times it.

The formula.

Every mode reduces to the side length first and then rebuilds the other five from it. From the perimeter, side = P ÷ 3. From the area, side = √(4A ÷ √3). From the height, side = 2h ÷ √3. From the inradius, side = 6r ÷ √3. From the circumradius, side = 3R ÷ √3. Once the side is known, the height is (√3 ÷ 2) × side, the area is (√3 ÷ 4) × side², the perimeter is 3 × side, the inradius is (√3 ÷ 6) × side and the circumradius is (√3 ÷ 3) × side, which are MathWorld's Equations 1, 2, 3 to 5 and 6 to 8 for the equilateral triangle.

Where the √3 comes from: drop the altitude from any vertex. Because the triangle is symmetric about that line, the foot lands exactly at the midpoint of the opposite side, splitting the shape into two congruent right triangles with hypotenuse a and short leg a ÷ 2. Euclid I.47 then gives the altitude as √(a² − (a ÷ 2)²) = √(3a² ÷ 4) = a√3 ÷ 2. That is the only irrational step on the page, and everything else inherits it.

Working the default all the way through with √3 = 1.7320508075688772935: the height is 10 × 1.7320508075688772935 ÷ 2 = 8.660254037844386, displayed as 8.6602540378. The area is 100 × 1.7320508075688772935 ÷ 4 = 43.30127018922193, displayed as 43.3012701892. The perimeter is 30 exactly. The inradius is 10 × 1.7320508075688772935 ÷ 6 = 2.886751345948129, displayed as 2.8867513459. The circumradius is 10 × 1.7320508075688772935 ÷ 3 = 5.773502691896258, displayed as 5.7735026919.

Three checks on those numbers, all of which the test suite asserts. First, ½ × side × height = ½ × 10 × 8.660254037844386 = 43.30127018922193, which is the area, as it must be. Second, the circumradius is exactly twice the inradius: 2 × 2.886751345948129 = 5.773502691896258. Third, the height is exactly three times the inradius: 3 × 2.886751345948129 = 8.660254037844386. An independent route to the same area is Heron's formula: with all three sides equal to 10 the semiperimeter is 15 and the area is √(15 × 5 × 5 × 5) = √1875 = 43.30127018922193 — the same figure from a formula that knows nothing about equilateral triangles.

Rounding stage: all arithmetic runs in Decimal at 40 significant digits and is rounded once, at the return boundary, to 10 decimal places. √3 is computed to that precision rather than typed as a literal, because the area is quadratic in the side and a rounded literal would show up in the ninth decimal place. The side is recovered at full precision and is never rounded before the other five outputs are derived from it. No trigonometric function is called anywhere in this calculator: sin 60°, cot(π/3) and csc(π/3) are all rational multiples of √3, so the exact algebraic forms are used and the double-precision error that affects general triangle solvers does not arise. Nothing on this page is sorted into bands or graded, so no threshold depends on display rounding.

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.