Solve any right triangle from two known values — legs, hypotenuse, or angles. Computes all sides, angles, area, perimeter, inradius, circumradius, and altitude. Includes Pythagorean triple presets.
A right triangle is a triangle with one angle equal to 90 degrees. It is the foundation of trigonometry and one of the most practical shapes in geometry, construction, navigation, and engineering. The Pythagorean theorem — a² + b² = c², where c is the hypotenuse — is arguably the most famous formula in all of mathematics.
Right triangles appear everywhere: roof pitches, ramp angles, screen diagonals, ladder safety calculations, surveying, and navigation. Whenever you need to find an unknown distance or angle given partial information, the right triangle is your tool.
This calculator is a complete right triangle solver. Enter any two of the six values — two legs, hypotenuse, or an acute angle — and it computes everything else: all three sides, both acute angles, area, perimeter, the inradius (radius of the inscribed circle), the circumradius (which is always half the hypotenuse), and the altitude to the hypotenuse.
Special right triangles like 30-60-90 and 45-45-90 have exact side ratios (1:√3:2 and 1:1:√2 respectively) that make hand calculation easy. Pythagorean triples — integer-sided right triangles like 3-4-5 and 5-12-13 — are invaluable in construction for verifying square corners. Presets for these and other common triangles are included for quick exploration.
Right triangles are the workhorse of geometry, surveying, construction, navigation, and trigonometry, so a fast solver is useful far beyond homework. This calculator lets you switch between common input combinations without rewriting formulas each time, which is especially helpful when one problem gives two legs, another gives a hypotenuse and angle, and a third needs the altitude or radius values derived from the solved triangle. It is built for quick verification, clean comparison of standard triples, and practical measurements like roof pitch, ladder reach, and diagonal layout checks.
Pythagorean theorem: a² + b² = c² Area: A = ½ab Perimeter: P = a + b + c Angle A: sin(A) = a/c → A = sin⁻¹(a/c) Angle B: B = 90° − A Inradius: r = (a + b − c) / 2 Circumradius: R = c / 2 Altitude to hypotenuse: h = ab / c
Result: Hypotenuse = 13, Area = 30, Perimeter = 30
In Two Legs mode, enter 5 and 12. The hypotenuse is c = √(5² + 12²) = √169 = 13, so the triangle is the classic 5-12-13 Pythagorean triple. Area is ½ × 5 × 12 = 30 cm², perimeter is 5 + 12 + 13 = 30 cm, the inradius is (5 + 12 − 13) / 2 = 2 cm, and the circumradius is 13 / 2 = 6.5 cm.
Any time a problem includes a horizontal component, a vertical component, and the straight-line distance between them, a right triangle is hiding underneath. That is why the same formulas apply to roof framing, wheelchair ramps, screen sizes, navigation bearings, and survey offsets. Learning to recognize that structure lets you turn messy real-world measurements into a clean and solvable model.
Different problems reveal different pairs of known values. In some settings you measure two legs directly, while in others you know the slope angle and one side, or the overall diagonal and an angle. A good solver should handle all of those cases without forcing you to convert everything manually first. This calculator mirrors that reality by letting you solve from the information you actually have rather than from a single rigid format.
Special right triangles are more than memorization exercises; they are powerful error checks. If a result is close to a 3-4-5, 5-12-13, 45-45-90, or 30-60-90 pattern, you can estimate whether the computed side lengths and angles make sense before trusting the exact decimal output. That habit is useful in class, on a job site, and anywhere you need a fast sanity check before cutting material or finalizing a drawing.
In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². Use this as a practical reminder before finalizing the result.
c = √(a² + b²). For example, legs 3 and 4 give c = √(9 + 16) = √25 = 5.
If you know leg a and hypotenuse c: b = √(c² − a²). For 5 and 13: b = √(169 − 25) = √144 = 12.
The radius of the largest circle that fits inside the triangle. For a right triangle: r = (a + b − c) / 2.
45-45-90 (isosceles right, sides 1:1:√2) and 30-60-90 (sides 1:√3:2). They have exact ratios, useful for hand calculations.
A set of three positive integers (a, b, c) where a² + b² = c². Examples: 3-4-5, 5-12-13, 8-15-17. They represent right triangles with integer side lengths.