Full right-triangle solver — give any two facts (two sides, or one side and one acute angle) and read every other measurement: sides, angles, area, perimeter, altitude, medians, inradius and circumradius.
Choose your input mode, type the two known values.
Area
Leg a
3
opposite A
Leg b
4
opposite B
Hypotenuse c
5
opposite 90°
Perimeter
12
a + b + c
Angle A · B
36.87° · 53.13°
sum to 90°
Altitude h
2.4
to hypotenuse
Inradius r
1
(a + b − c)/2
Circumradius R
2.5
c / 2
| Quantity | Formula | Notes |
|---|---|---|
| Hypotenuse | c = √(a² + b²) | Pythagorean theorem |
| Area | ½ · a · b | legs as base & height |
| Altitude (to hyp) | h = a · b / c | geometric mean |
| Inradius | r = (a + b − c) / 2 | = Area / s |
| Circumradius | R = c / 2 | Thales' theorem |
| Median to hyp | m_c = c / 2 | equals R |
The Method
A right triangle has one angle equal to 90°, so the other two angles must sum to 90°. That single constraint, combined with the Pythagorean theorem a² + b² = c² and the basic trigonometric ratios sin A = a/c, cos A = b/c, tan A = a/b, is enough to recover every measurement from just two facts. Once you have all three sides, the area is ½·a·b, the altitude to the hypotenuse is h = a·b/c, the inradius is r = (a+b−c)/2, and the circumradius is R = c/2 (the hypotenuse is the diameter of the circumscribed circle — Thales' theorem).
Working for your triangle
About This Tool
A right triangle calculator takes any two known measurements of a right-angled triangle and solves the rest. Where a basic Pythagorean tool only handles two sides at a time, this calculator accepts seven different input modes — combinations of sides and acute angles — and returns the full set of derived quantities: all three sides, both non-right angles, the area, the perimeter, the altitude to the hypotenuse, the inradius, and the circumradius.
Right triangles are the workhorses of geometry and trigonometry. They appear in construction (roof pitch, stair stringers), navigation (resolving distances into north/east components), physics (decomposing forces into perpendicular axes), surveying, and computer graphics. The trigonometric ratios sin, cos and tan are defined on right triangles, then extended to all angles via the unit circle.
Mathematically, every right-triangle problem reduces to the Pythagorean theorem plus the three basic trig ratios. Once you know two facts, the right-angle constraint plus these formulas determine everything else. The tool walks you through each substitution and verifies the result against multiple identities — handy for spotting input errors.
Use this free right triangle calculator for geometry homework, trigonometry exams, DIY projects, surveying calculations, or anywhere you need the full solution of a right triangle from limited information. All computation runs locally — no sign-up, no tracking.
7 Input Modes
Any two facts work — two sides, side + angle, etc.
Full Solution
Sides, angles, area, perimeter, altitude, inradius, circumradius — all returned.
Labelled Diagram
Scaled triangle with altitude and inscribed circle drawn to your inputs.
Step-by-Step Working
Pythagoras and trig substitutions laid out — ready for homework write-up.
100% Free & Private
No account, no tracking — every calculation runs locally in your browser.
Triple Detection
Flags classic Pythagorean triples (3-4-5, 5-12-13, 8-15-17, …).
Choose what you know, type two values, read everything else.
Choose from the dropdown — two legs, leg + hypotenuse, or side + angle. Only the two relevant input fields are shown.
Type each value in the matching field. Use any positive number for sides; angles are in degrees between 0 and 90.
The summary card headlines the area; the stat tiles below show every side (a, b, c) and the perimeter.
The two acute angles (A, B) are reported in degrees and always sum to 90° — a built-in sanity check.
A scaled diagram shows your triangle with the altitude to the hypotenuse and the inscribed circle overlaid — useful for visualising geometry.
Open the formula card for every substitution — Pythagoras, trig ratios, altitude, inradius — perfect for homework write-up.
Everything you need to know about right triangles — definitions, formulas, and the common pitfalls.
A right triangle is a triangle with exactly one interior angle of 90°. The side opposite the right angle is the hypotenuse (always the longest side), and the other two sides are the legs. The two non-right angles are always acute and sum to 90°, since all interior angles of any triangle add to 180°.
Any two independent facts are enough. With two sides, use the Pythagorean theorem to find the third. With one side and one acute angle, use the three trig ratios: sin A = a/c, cos A = b/c, tan A = a/b. Once all three sides are known, area, altitude, inradius and circumradius follow from standard formulas.
The altitude to the hypotenuse is the perpendicular distance from the right-angle vertex to the hypotenuse. Formula: h = a · b / c. It splits the original triangle into two smaller right triangles, both similar to the original — a useful result that yields several geometric mean identities (e.g. h² = p · q, where p and q are the two pieces of the hypotenuse).
The inradius (r) is the radius of the inscribed circle — the largest circle that fits inside the triangle, tangent to all three sides. For a right triangle there's an unusually neat formula: r = (a + b − c) / 2. More generally, r = Area / s, where s = (a+b+c)/2 is the semi-perimeter.
The circumradius (R) is the radius of the circumscribed circle, which passes through all three vertices. For a right triangle, it's simply R = c / 2 — the hypotenuse is the diameter of the circumscribed circle. This is Thales' theorem: an inscribed angle subtending a diameter is always 90°.
For angle A opposite side a, with hypotenuse c: sin A = opposite / hypotenuse = a/c; cos A = adjacent / hypotenuse = b/c; tan A = opposite / adjacent = a/b. Mnemonic: SOH-CAH-TOA. The remaining ratios (cosec, sec, cot) are their reciprocals. These definitions are extended to all angles via the unit circle.
All interior angles of a triangle sum to 180°. In a right triangle one angle is already 90°, so the remaining two must sum to 180° − 90° = 90°. Two angles that sum to 90° are called complementary — a nice consequence: sin A = cos B and cos A = sin B whenever A + B = 90°.
Area = (1/2) · a · b — half the product of the two legs. The legs meet at the right angle and serve directly as base and height, with no perpendicular projection needed. Alternative formula: Area = (1/2) · c · h, where h is the altitude to the hypotenuse. Both give the same answer.
Some inputs don't correspond to a valid right triangle. For example, with two sides where the hypotenuse equals or is smaller than a leg, no real triangle exists. Or an angle ≥ 90° (only acute angles allowed besides the fixed right angle). The calculator flags these cases and tells you which constraint is violated.
Two stand out. The 45-45-90 triangle (isoceles right): legs equal, hypotenuse = leg·√2. The 30-60-90 triangle: sides in ratio 1 : √3 : 2 (short leg : long leg : hypotenuse). Both come from cutting common shapes (square diagonal, equilateral triangle altitude) and are memorised exactly — no calculator needed.
All calculations use IEEE 754 double-precision floating-point arithmetic via JavaScript's Math.sqrt, Math.sin, Math.cos, Math.tan and the inverse trig functions — accurate to about 15–17 significant digits. Results are reported to 4 decimal places, which is far more than any physical measurement justifies.
Everywhere. Construction (3-4-5 to check square corners; rafter and stair calculations). Surveying (triangulating distances). Navigation (resolving heading into north/east components). Physics (decomposing a force vector into perpendicular components). Computer graphics (every pixel is positioned via 2D coordinates and uses right-triangle distance and angle math). Trigonometry itself was invented to solve right triangles.