Right Triangle Calculator
Solve a right triangle from one angle and the hypotenuse.
What this calculator does
Solves a right triangle completely — both remaining side lengths and the other angle — from just one known angle and the hypotenuse, using standard trigonometric ratios.
Who this is for
Students checking geometry or trigonometry homework, anyone working on a construction, carpentry, or design problem involving right-angle measurements, or people double-checking a manual trig calculation.
Methodology
Given one angle and the hypotenuse of a right triangle, the other two sides follow directly from the basic trig ratios: Opposite = Hypotenuse × sin(angle), and Adjacent = Hypotenuse × cos(angle). The remaining angle is simply 90° minus the known angle, since all three angles in any triangle sum to 180° and one is already 90°.
Worked example
A 30° angle with a hypotenuse of 10: Opposite = 10 × sin(30°) = 10 × 0.5 = 5. Adjacent = 10 × cos(30°) = 10 × 0.866 = 8.66. The remaining angle = 90° − 30° = 60°. Checking with the Pythagorean theorem: 5² + 8.66² = 25 + 75 = 100 = 10², confirming the result.
Interpretation
The three core ratios — sine, cosine, and tangent — each relate two of the triangle's three sides to the chosen angle. Sine compares the opposite side to the hypotenuse, cosine compares the adjacent side to the hypotenuse, and tangent compares the opposite side directly to the adjacent side (without involving the hypotenuse at all). These ratios are constant for a given angle, regardless of the triangle's actual size — which is exactly why they're useful for solving triangles of any scale.
Triangle sides compared
Run the calculator above to see the three side lengths compared.
Common mistakes
- Mixing up opposite and adjacent. Which side is "opposite" versus "adjacent" depends entirely on which angle you're measuring from — they swap if you switch to the other non-right angle.
- Using degrees when a function expects radians (or vice versa). Most calculators and programming languages default to radians for trig functions — always confirm which mode you're in.
- Forgetting the angle sum check. If your two non-right angles don't add up to 90°, something in the setup is wrong.
- Not verifying with the Pythagorean theorem. Once you have all three sides, checking that a² + b² = c² is a quick way to catch calculation errors before relying on the result.
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Frequently Asked Questions
How can I check my answer is correct?
Use the Pythagorean theorem (a² + b² = c²) once you have all three sides — if the equation balances, your calculation is correct. You can also check that your two non-right angles sum to exactly 90°.
What is SOH-CAH-TOA?
A memory aid for the three basic trig ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Why do the two non-right angles always add up to 90 degrees?
All three angles in any triangle sum to 180 degrees. Since one angle is already 90 degrees, the other two must together make up the remaining 90 degrees.
What is the Pythagorean theorem used for here?
It relates the three sides of a right triangle (a² + b² = c²), used here to double-check the third side once two are known.