30-60-90 Triangle Calculator

Side ratios

Introduce s = side opposite 30°. Then by trigonometry or by the equilateral-bisection argument:

s : s√(3) : 2s \;=\; 1 : √(3) : 2

Derivation from an equilateral triangle

An equilateral triangle has three sides of length a and three angles of 60°. Drop a perpendicular from one vertex to the opposite side: it bisects that side (length a/2) and creates two 30-60-90 triangles with sides s = a/2 (opposite 30°), long leg a√3/2, and hypotenuse a.

Area and perimeter

A 30-60-90 triangle has legs of lengths s and s√3, so:

A = \tfrac12 · s · s√(3) = \tfrac√(3)2 \, s^2, \qquad P = s + s√(3) + 2s = s\,(3 + √(3))

Comparing to 45-45-90 and scalene

Among the three Pythagorean triples in this family (3-4-5, 5-12-13, 8-15-17, …) and the two "infamous" special right triangles (45-45-90 and 30-60-90), the 30-60-90 is the building block of 60°/120° coordinate systems including the cubic lattice and hexagonal cells.

Deriving the 1 : √3 : 2 Ratio

Take an equilateral triangle of side 2 and drop a perpendicular from one vertex to the opposite side. That altitude bisects both the 60° apex angle and the base, producing two identical 30-60-90 triangles.

Each has a hypotenuse of 2 (the original side), a short leg of 1 (half the bisected base), and by Pythagoras a long leg of √(4-1) = √(3). This construction proves the ratio without any trigonometry and fixes which side goes where: the short leg faces the 30° angle, the long leg faces the 60° angle, and the hypotenuse faces the right angle. With √(3) ≈ 1.7320508, a short leg of 5 gives a long leg of about 8.66 and a hypotenuse of 10.

a : a√(3) : 2a, \qquad \sin 30° = \tfrac12, \quad \sin 60° = (√(3))/(2), \quad \tan 30° = (1)/(√(3))

Converting Between Any Two Sides

All six conversions follow from the ratio. From the short leg: multiply by √(3) for the long leg, by 2 for the hypotenuse. From the long leg: divide by √(3) for the short leg, multiply by 2/√(3) for the hypotenuse. From the hypotenuse: halve it for the short leg, multiply by √(3)/2 for the long leg.

The reliable way to avoid mixing these up is to always route through the short leg, since it is the "1" of the ratio. Given anything, reduce it to the short leg first, then scale out to whatever you need. A sanity check that catches most slips: the sides must order as short leg < long leg < hypotenuse, with the long leg roughly 1.73 times the short and the hypotenuse exactly twice it.

a = (h)/(2) = (b)/(√(3)), \qquad b = a√(3) = (h√(3))/(2), \qquad h = 2a = (2b)/(√(3))

Area and Where the Shape Appears

The two legs are perpendicular, so the area is A = \tfrac12ab = \tfrac√(3)2a^2, and the perimeter is a(3 + √(3)) ≈ 4.732a. Running the equilateral construction in reverse gives that triangle's altitude as \tfrac√(3)2s and its area as \tfrac√(3)4s^2 — both standard results that come free from this triangle.

It shows up constantly. A regular hexagon is six equilateral triangles, so hexagonal nuts, honeycomb panels and tiling layouts all reduce to 30-60-90 arithmetic; the across-flats and across-corners dimensions of a hex nut differ by exactly 2/√(3). Isometric drawing uses 30° axes for the same reason. In three-phase electrical work the √(3) between line and phase voltage is this triangle in phasor form, and roof pitches, ramp gradients and staircase stringers at 30° are laid out directly from the ratio.

A = (√(3))/(2)a^2 = (h^2√(3))/(8)

Worked Example

Inputs

• shortLeg: 1

Result: longLeg: 1.7321 hypotenuse: 2 area: 0.866 perimeter: 4.7321

Explanation

With s = 1, the side opposite 60° is √3 ≈ 1.7321, the hypotenuse is 2, the area is (√3/2) ≈ 0.8660, and the perimeter is 3 + √3 ≈ 4.7321.

Second Scenario

Inputs

• shortLeg: 2.25

Result: longLeg: 1.7321 hypotenuse: 2 area: 0.866 perimeter: 4.7321

Explanation

This scenario uses different inputs (shortLeg = 2.25) to show how changing one variable affects the 30-60-90 triangle result. Run the calculator above with these values to get the exact updated output with step-by-step work.