About

A right triangle is fully determined by two independent measurements. Given any two of the three sides (a, b, c) or one side and one acute angle, all remaining elements can be recovered through the Pythagorean relation a² + b² = c² and inverse trigonometric functions. Errors in angle computation propagate into structural layouts, CNC machining offsets, and surveying baselines. A 0.5° miscalculation on a 10m rafter produces an offset of approximately 87mm at the endpoint.

This calculator solves all six unknowns (α, β, a, b, c, plus area and perimeter) from any valid pair of inputs. It assumes Euclidean geometry on a flat plane. Results are limited by IEEE 754 double-precision floating-point arithmetic, yielding approximately 15 significant digits. For geodetic triangles on Earth’s surface where sides exceed 100km, spherical excess becomes non-negligible and this tool should not be used.

Formulas

All computations derive from the Pythagorean theorem and the definitions of sine, cosine, and tangent in a right triangle with legs a (opposite α), b (opposite β), and hypotenuse c.

a² + b² = c²

Angle recovery from two known sides:

α = atan(ab)

β = 90° − α

Side recovery from one side and one angle:

a = c ⋅ sin(α)

b = c ⋅ cos(α)

Derived quantities:

A = 12 ⋅ a ⋅ b

P = a + b + c

h = a ⋅ bc

Where a = leg opposite angle α, b = leg opposite angle β, c = hypotenuse, A = area, P = perimeter, h = altitude to hypotenuse.

Reference Data

Triangle Name	Angle α	Angle β	Side Ratio a : b : c	Exact Values	Common Use
45-45-90	45°	45°	1 : 1 : √2	sin(45°) = √22	Diagonal of a square
30-60-90	30°	60°	1 : √3 : 2	sin(30°) = 12	Equilateral triangle bisection
3-4-5	36.87°	53.13°	3 : 4 : 5	Smallest Pythagorean triple	Construction layout verification
5-12-13	22.62°	67.38°	5 : 12 : 13	Pythagorean triple	Roof pitch calculations
8-15-17	28.07°	61.93°	8 : 15 : 17	Pythagorean triple	Framing and carpentry
7-24-25	16.26°	73.74°	7 : 24 : 25	Pythagorean triple	Surveying baselines
9-40-41	12.68°	77.32°	9 : 40 : 41	Pythagorean triple	Long-span structures
20-21-29	43.60°	46.40°	20 : 21 : 29	Pythagorean triple	Near-isosceles right triangle
11-60-61	10.39°	79.61°	11 : 60 : 61	Pythagorean triple	Shallow-angle ramps
6-8-10	36.87°	53.13°	6 : 8 : 10	Scaled 3-4-5	Double-scale layout
Approx 1:2 slope	26.57°	63.43°	1 : 2 : √5	tan(α) = 0.5	Common roof pitch (6:12)
Approx 1:3 slope	18.43°	71.57°	1 : 3 : √10	tan(α) = 0.333	Gentle slope / ADA ramp
Key Trigonometric Values
Reference	Angle		sin	cos	tan
-	0°		0	1	0
-	15°		0.2588	0.9659	0.2679
-	30°		0.5	0.8660	0.5774
-	45°		0.7071	0.7071	1
-	60°		0.8660	0.5	1.7321
-	75°		0.9659	0.2588	3.7321
-	90°		1	0	∞

Frequently Asked Questions

Any two independent measurements suffice: two sides, or one side plus one acute angle. Two angles alone (other than the fixed 90°) cannot determine the triangle because they only fix the shape, not the scale. For example, providing leg a = 3 and leg b = 4 yields hypotenuse c = 5, α ≈ 36.87°, and β ≈ 53.13°.

IEEE 754 double-precision provides roughly 15 significant decimal digits. For side lengths below 1e-8 or above 1e+15, accumulated rounding in atan and sqrt may reduce output accuracy to 10 - 12 digits. The calculator rounds displayed results to 4 decimal places, which masks sub-ulp errors in all practical scenarios.

No. A right triangle's acute angles must satisfy 0° < α < 90°. An angle of 0° collapses the triangle into a line segment, and 90° would create a second right angle, violating the angle-sum constraint of 180°. The calculator rejects both and displays a validation error.

The altitude h from the right-angle vertex to the hypotenuse equals a · bc. It appears in structural engineering (maximum beam depth), optics (lens sag calculations), and the geometric mean relations. The two segments of the hypotenuse created by this altitude satisfy a² = c · p and b² = c · q, where p and q are the segments.

The calculator displays all three units simultaneously. The conversion factors are: 1° = π180 rad ≈ 0.01745 rad, and 1° = 109 grad. Radians are standard in most programming languages and physics equations. Gradians (also called gon) are used in European surveying instruments.

If you enter two sides labeled as legs, there is no constraint violation because any two positive numbers can be legs of a right triangle. However, if you enter a leg value greater than or equal to the hypotenuse, the calculator rejects the input because the hypotenuse must always be the longest side. Specifically, a < c and b < c must hold.