About

A 30-60-90 triangle is one of two special right triangles in Euclidean geometry (the other being the 45-45-90). Its internal angles measure 30°, 60°, and 90°, producing a fixed side ratio of 1 : √3 : 2. The short leg a sits opposite the 30° angle. The long leg b sits opposite 60°. The hypotenuse c sits opposite the right angle. Misidentifying which side is which is the most common source of error in structural and mechanical calculations that depend on this triangle. This calculator eliminates that risk by deriving all measurements from a single known side.

The 30-60-90 triangle appears in equilateral triangle bisection, hexagonal geometry, roof pitch framing at 30° slope, and crystallography. The ratios are exact and irrational. Rounding √3 to 1.73 introduces a 0.02% truncation error. This tool preserves full floating-point precision internally and rounds output to a user-selected number of decimal places. Note: all calculations assume a flat Euclidean plane. The ratios do not hold on curved surfaces.

Formulas

All side relationships in a 30-60-90 triangle derive from its fixed ratio. Given the short leg a (opposite the 30° angle), every other measurement follows algebraically.

b = a ⋅ √3

c = 2a

Area is computed as half the product of the two legs:

A = 12 ⋅ a ⋅ b = √34 ⋅ a²

Perimeter sums all three sides:

P = a + b + c = a(1 + √3 + 2) = a(3 + √3)

The altitude from the right-angle vertex to the hypotenuse:

h = 2Ac = a √32

Incircle radius for a right triangle with legs a, b and hypotenuse c:

r = a + b − c2

Circumradius for any right triangle:

R = c2

Where a = short leg (opposite 30°), b = long leg (opposite 60°), c = hypotenuse (opposite 90°), A = area, P = perimeter, h = altitude to hypotenuse, r = inradius, R = circumradius.

Reference Data

Property	Value / Ratio	Notes
Angles	30°, 60°, 90°	Fixed. Sum = 180°
Side ratio	1 : √3 : 2	Short leg : Long leg : Hypotenuse
Short leg a	a = c ÷ 2	Opposite 30°
Long leg b	b = a ⋅ √3	Opposite 60°
Hypotenuse c	c = 2a	Opposite 90°
sin(30°)	12 = 0.5	Exact value
cos(30°)	√32 ≈ 0.8660	Exact value
tan(30°)	1√3 ≈ 0.5774	Exact value
sin(60°)	√32 ≈ 0.8660	Exact value
cos(60°)	12 = 0.5	Exact value
tan(60°)	√3 ≈ 1.7321	Exact value
Area	A = √34 a²	In terms of short leg
Perimeter	P = a(3 + √3)	In terms of short leg
Height to hypotenuse h	h = a √32	Altitude from right angle to hypotenuse
Inradius r	r = a(3 + √3 − 2√3)2	Inscribed circle radius; simplified: r = a(3 − √3)2
Circumradius R	R = a	For right triangle: R = c2 = a
Median to hypotenuse	m_c = a	Equals circumradius for right triangles
Origin	Half of equilateral triangle	Bisecting any equilateral triangle yields two 30-60-90 triangles
30° in radians	π6 ≈ 0.5236	Exact conversion
60° in radians	π3 ≈ 1.0472	Exact conversion
90° in radians	π2 ≈ 1.5708	Exact conversion

Frequently Asked Questions

The short leg is always opposite the smallest angle (30°). The long leg is opposite 60°. The hypotenuse is opposite 90° and is always the longest side. If you know the hypotenuse c, the short leg is c ÷ 2 and the long leg is c ⋅ √3 ÷ 2.

An equilateral triangle has three 60° angles. Drawing an altitude from one vertex to the opposite side bisects the vertex angle into two 30° angles and creates a right angle at the base. The altitude becomes the long leg, half the base becomes the short leg, and the original side becomes the hypotenuse. This geometric fact is why the ratio 1 : √3 : 2 is exact and not an approximation.

The value √3 is irrational (1.7320508...). Truncating to 1.73 gives a relative error of about 0.024%. For a side length of 10 m, that is a 2.4 mm discrepancy. This calculator uses IEEE 754 double-precision floats internally (~15 significant digits), so the computational error is negligible for all practical applications.

This calculator requires one side as input. However, you can reverse-engineer a side from the area: a = √4A√3. From perimeter: a = P3 + √3. Compute the short leg first, then enter it into the calculator.

The altitude h from the right-angle vertex to the hypotenuse can be computed via area equivalence: h = a ⋅ bc. For a 30-60-90 triangle this simplifies to a √32. This altitude also splits the hypotenuse into two segments of length a2 and 3a2.

Roof framing at a 30° pitch uses this triangle directly. Hexagonal bolt heads and nuts are dimensioned using 30-60-90 geometry. Crystal lattice structures (hexagonal close-packed) rely on these ratios. Surveying slope corrections at 30° or 60° grades, staircase design with 30° incline, and even clock-face geometry (each hour mark subtends 30°) all use these exact ratios.