30-60-90 Triangle Calculator

Background.

Fold an equilateral triangle in half, crease running from one corner to the middle of the opposite side, and each half is a 30-60-90 triangle. Everything about the shape follows from that fold. The crease meets the base at a right angle; it splits the 60° corner it starts from into two 30° angles; the base is cut into two equal halves; and the slanted edge of each half is still a full side of the original triangle. So the hypotenuse is exactly twice the short leg, and Pythagoras gives the third side as √(4s² − s²) = s√3.

Those fixed proportions, 1 : √3 : 2, mean one measurement is enough. A general right triangle needs two numbers before its other sides can be found, but here the angles are already known, so give the short leg, the long leg, the hypotenuse, the area or the perimeter and the rest of the triangle follows, along with the altitude dropped from the right angle onto the hypotenuse.

Two facts do most of the mental arithmetic. Doubling the short leg gives the hypotenuse, exactly, with no square root involved. Multiplying the short leg by about 1.7321 gives the long leg, which is the side people usually need a calculator for. Because √3 never terminates, the long leg is never a whole number when the short leg is one, and vice versa.

The shape turns up wherever hexagons and equilateral triangles do, from hex nuts to floor tiles. Isometric drawings use axes at 30° to the horizontal, the 30/60 set square is this triangle in plastic, and its side ratios give the exact values sin 30° = 1/2 and tan 60° = √3 that trigonometry students memorise.

What is 30-60-90 triangle calculator?

A 30-60-90 triangle is a right triangle whose two acute angles are 30° and 60°. Its sides are always in the ratio 1 : √3 : 2: the short leg lies opposite the 30° angle, the long leg opposite the 60° angle, and the hypotenuse opposite the right angle. Every triangle with these angles is a scaled copy of every other, so naming any one length, its area or its perimeter fixes the whole triangle. MathWorld states the same ratio through trigonometry: a hypotenuse a has legs a·sin 30° = ½a and a·sin 60° = ½a√3.

How to use this calculator.

1. 01Pick the measurement you have from the list: short leg, long leg, hypotenuse, area or perimeter.
2. 02Make sure you have named the side correctly. The short leg is opposite the 30° angle and is always the shortest side; the hypotenuse is opposite the right angle and is always the longest. If a drawing is rotated, find the right-angle mark first.
3. 03Enter the value in whatever unit you measured in, inches or millimetres alike; the sides are returned in it, and the area in its square.
4. 04Check that the hypotenuse is exactly twice the short leg. If you expected that and it is not what you see, you probably selected the wrong side.
5. 05If your angles are not exactly 30° and 60° — a cut that came out at 31° and 59°, say — use a general triangle solver with the measurements you actually have.

The formula.

Every mode first finds the short leg s and then rebuilds the triangle from it: long leg = s√3, hypotenuse = 2s, area = ½ × s × s√3 = (√3 ÷ 2)s², perimeter = s + s√3 + 2s = s(3 + √3), and altitude to the hypotenuse = (s × s√3) ÷ 2s = (s√3) ÷ 2.

Going backwards, a long leg L gives s = L ÷ √3, a hypotenuse H gives s = H ÷ 2, an area A gives s = √(2A ÷ √3), and a perimeter P gives s = P ÷ (3 + √3), where 3 + √3 is about 4.7321.

The area coefficient is where mistakes cluster. For this triangle it is √3 ÷ 2, about 0.866; for an equilateral triangle of side a it is √3 ÷ 4. Both appear because the 30-60-90 triangle is half of an equilateral whose side is 2s, not s: half of (√3 ÷ 4)(2s)² is (√3 ÷ 2)s². Reaching for the equilateral formula with the short leg plugged in gives exactly half the true area.

The derivation needs no trigonometry: Euclid's Elements constructs the equilateral triangle in Book I, Proposition 1, and the Pythagorean theorem of Proposition 47 gives the long leg. MathWorld's sine-based statement agrees with it.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Wolfram Research, MathWorld — '30-60-90 Triangle' (live revision retrieved 2026-07-29). States that for a hypotenuse of length a the two legs are a·sin(60°) = ½a√3 and a·sin(30°) = ½a, which is the 1 : √3 : 2 ratio this page uses. Open access.
2. [2]Euclid, Elements, Book I, Proposition 1 — 'To construct an equilateral triangle on a given finite straight line.' The parent figure of the bisection derivation. David E. Joyce web edition of the Heath translation, Clark University; statement confirmed verbatim on retrieval 2026-07-29. Open access.
3. [3]Euclid, Elements, Book I, Proposition 47 — 'In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.' Applied to the bisected equilateral triangle it gives the long leg as √(4s² − s²) = s√3, an entirely geometric derivation of the ratio that uses no trigonometry and serves as the independent check on this page's constant. Joyce web edition; confirmed verbatim on retrieval 2026-07-29. Open access.
4. [4]NIST Digital Library of Mathematical Functions, §4.14 — definitions and periodicity of the circular functions, the basis of the trigonometric statement of the side ratios (sin 30° = 1/2, sin 60° = √3/2). Open access.