Right Triangle Calculator
Solve a right triangle completely — find all sides and angles from any two known values using the Pythagorean theorem and trigonometry.
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How to use this calculator
Given any two sides or one side and one acute angle, all remaining parts of a right triangle can be determined using the Pythagorean theorem and the trigonometric ratios.
- 1
Select which two values you know from the dropdown (e.g., "Legs a and b").
- 2
Enter the first value in the "First value" field and the second in "Second value".
- 3
All remaining sides, angles, area, and perimeter are calculated instantly.
- 4
Angles are in degrees; all sides share the same unit you entered.
Frequently asked questions
What is the Pythagorean theorem?
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (longest side, opposite the right angle) equals the sum of squares of the other two legs: a² + b² = c². It is one of the most fundamental results in Euclidean geometry.
How do I find a missing angle in a right triangle?
If you know two sides, use inverse trig: angle A = arcsin(opposite/hypotenuse) = arccos(adjacent/hypotenuse) = arctan(opposite/adjacent). Since the right angle is always 90°, the third angle is simply 90° minus the other known angle.
What are common Pythagorean triples?
Pythagorean triples are whole-number solutions: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and any multiples thereof (e.g. 6-8-10). They appear frequently in construction, carpentry, and standardized tests.
Can I use this calculator for non-right triangles?
No — this calculator only handles right triangles (one 90° angle). For other triangles, use the Law of Cosines or Law of Sines calculators, which handle any triangle given sufficient information.
Right triangle calculator — solve any right triangle instantly
Understanding the right triangle
A right triangle has one interior angle of exactly 90°. The side opposite the right angle is called the hypotenuse — always the longest side. The other two sides are legs (or catheti). Knowing any two of the six measurements (three sides, three angles) completely determines the triangle. This calculator handles all input combinations: two legs, leg plus hypotenuse, or one leg plus one acute angle.
Trigonometric ratios: SOH-CAH-TOA
The memory aid SOH-CAH-TOA summarizes the three primary trig ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. These ratios remain constant for a given angle regardless of triangle size, making them the bridge between angles and side lengths. Inverse functions (arcsin, arccos, arctan) recover angles from side ratios.
Real-world applications
Right triangles underlie navigation (finding distances via GPS triangulation), construction (roof pitch, stair risers, ramp slopes), physics (resolving force vectors), and surveying. A 3-4-5 triangle is the classic carpenter's square check — if a corner's sides measure 3 and 4 units, the diagonal should be exactly 5, confirming a 90° angle. Engineers use the Pythagorean theorem constantly when resolving horizontal and vertical components of forces and displacements.
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