Free right triangle calculator
Solve any right triangle from two known sides or angles — enter your values to find all sides, angles, and area, updated live, as you type.
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Pythagorean theorem: a² + b² = c². Angles A and B are in degrees. C = 90°.
Results are estimates. Consult a professional.
How the right triangle calculator works
A right triangle contains one 90° angle. Given any two sides, the Pythagorean theorem finds the third. Given a side and an angle, the trigonometric ratios (sin, cos, tan) determine the remaining sides and angles. The three angles always sum to 180°.
Worked example: legs a = 6, b = 8
The classic 3-4-5 scaled to 6-8-10 right triangle. Starting from two legs, we find the hypotenuse using the Pythagorean theorem, then derive both acute angles using inverse tangent.
Common right triangle types
Certain right triangles appear so often in geometry and construction that their side ratios and angles are worth memorising. Pythagorean triples always give integer side lengths.
| Triangle type | Angles | Side ratio |
|---|---|---|
| 3-4-5 triple | ≈ 36.87° / 53.13° / 90° | 3 : 4 : 5 |
| 5-12-13 triple | ≈ 22.62° / 67.38° / 90° | 5 : 12 : 13 |
| 8-15-17 triple | ≈ 28.07° / 61.93° / 90° | 8 : 15 : 17 |
| 45-45-90 | 45° / 45° / 90° | 1 : 1 : √2 ≈ 1.414 |
| 30-60-90 | 30° / 60° / 90° | 1 : √3 : 2 ≈ 1 : 1.732 : 2 |
Source: standard Euclidean geometry and Pythagorean triples
Tips for right triangle problems
Right triangles underpin construction, navigation, surveying, and physics. Getting the formula and angle interpretation right is essential for accurate results.
- SOHCAHTOA as a mnemonic — Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. Write it down until it's automatic.
- Hypotenuse is always the longest side — it sits opposite the 90° angle. If your computed hypotenuse is shorter than either leg, check whether you squared instead of taking the square root.
- Use arctan to find an angle from two sides — if you know opposite and adjacent lengths, the angle is arctan(opposite/adjacent). Most calculators label this tan⁻¹.
- Confirm your angle mode — scientific calculators default to radians or degrees depending on settings. Always verify you are in degree mode for angle outputs expressed in degrees.
- Scale Pythagorean triples for easy construction — multiply 3-4-5 by any factor (e.g. × 4 = 12-16-20) to verify a right angle on-site without a protractor.
Accuracy and limitations
Side lengths are computed to full IEEE 754 double-precision and displayed rounded to the configured decimal places (default: 4). Angles derived from inverse trigonometric functions (arcsin, arccos, arctan) are accurate to approximately 13–14 significant digits. Inputs of zero or negative side lengths are invalid and will produce an error. When the input triangle is very obtuse (nearly 90° at one of the acute angles), the complementary angle approaches 0° and small input errors produce proportionally larger angle errors — standard behaviour of arctangent near 0.
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About this calculator
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