Right Triangle Calculator
Adjust the inputs below. Results update as you type.
How it works
Given hypotenuse c and angle A: opposite = c×sin(A), adjacent = c×cos(A). Given two legs a and b: hypotenuse = √(a²+b²), angle A = arctan(a/b). Angles always sum to 180°; right angle is 90°, so A+B = 90°. Area = ½ × base × height = ½ab. SOH-CAH-TOA: sin=opposite/hypotenuse, cos=adjacent/hypotenuse, tan=opposite/adjacent. Essential for surveying, construction, and navigation.
Input guidance
- Check units and decimal placement before running calculations.
- For percentage and ratio tasks, define the baseline value clearly.
- Use rounded output carefully and keep full precision for intermediate steps.
The formula
A right triangle combines the Pythagorean theorem (a² + b² = c²) with trigonometry: sin, cos, and tan relate the angles to the side ratios.
Worked example
With legs 3 and 4, the hypotenuse is 5, and the angle opposite the side of length 3 is arctan(3/4) ≈ 36.9°.
More examples to test
- Baseline example: solve with clean numbers first to verify formula direction.
- Edge-case example: test near-zero or boundary values to avoid interpretation errors.
How to interpret results
Use outputs as quick checks, then verify with step-by-step work for graded or high-stakes use.
When this can be inaccurate
Errors usually come from unit mismatch, wrong baseline assumptions, or rounding too early in multistep problems.
Change history
- July 2026: Quality-reviewed for publication with formula checks and explanatory copy updates.
Site-wide corrections also appear on the corrections log.
Build math and logic skills with guided lessons
Educational platform recommendation.
Frequently asked questions
What do I need to solve a right triangle?+
Any two pieces beyond the right angle — two sides, or one side and one acute angle — are enough to find everything else.
How does SOH-CAH-TOA help?+
It recalls the ratios: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent.
What do the angles sum to?+
The two acute angles add to 90°, since the right angle already accounts for 90° of the 180° total.
Why do my classroom and calculator results differ?+
Differences usually come from rounding conventions, order-of-operations handling, or different baseline assumptions.
What is the safest way to validate an answer?+
Recompute with units shown and check against a second method or manual arithmetic for critical problems.