How to use it
- In Find a missing side, type the two sides you know and leave the third box empty. Legs a and b meet at the right angle; the hypotenuse c is the longest side, opposite the right angle.
- The answer comes in simplest radical form (for legs 1 and 1, c = √2) and as a decimal, with the working, both acute angles, the area, the perimeter and the height to the hypotenuse.
- In Is it a right triangle?, type all three sides to see whether they make a right, acute or obtuse triangle, or no triangle at all.
The theorem and the two ways to use it
In a right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides: a² + b² = c². To find the hypotenuse, add the squares and take the square root: legs 3 and 4 give √(9 + 16) = √25 = 5. To find a leg, subtract instead: hypotenuse 13 and leg 12 give √(169 − 144) = √25 = 5. A leg can never come out longer than the hypotenuse, so a “hypotenuse” shorter than the known leg gets a message rather than an answer.
Simplest radical form
Most answers are not whole numbers. Legs 2 and 4 give c = √20, which simplifies to 2√5 because 20 = 4 × 5 and √4 = 2. The calculator always pulls every square factor out of the root so the number left under it has none. Fractions and decimals work too: legs 1/2 and 1/3 give √(1/4 + 1/9) = √(13/36) = (1/6)√13, and legs 2.5 and 6 give exactly 13/2.
Is it a right triangle?
The converse of the theorem works as a test. Take the longest side as c: if a² + b² equals c², the angle opposite c is exactly 90°; if it is larger the triangle is acute, and if it is smaller the triangle is obtuse. Whole-number right triangles such as (3, 4, 5), (5, 12, 13) and (8, 15, 17) are called Pythagorean triples, and any multiple of one, such as (6, 8, 10), is one too. Before any of that, three lengths only make a triangle if the two shorter ones add up to more than the longest.
What we tested
On October 15, 2026 we ran the page’s own functions on 3,000 random right-triangle problems (whole numbers, decimals, fractions and mixed numbers) and checked them with the SymPy computer-algebra library: 1,037 hypotenuses and 983 legs, 2,010 of them irrational, all equal to the exact square root with nothing square left under the radical, and the 980 impossible ones (a hypotenuse no longer than the leg) refused. 3,000 sets of three sides were classified exactly as an exact fraction comparison does: 1,000 right, 84 acute, 171 obtuse and 1,745 that cannot form a triangle. In Chromium: legs 3 and 4 gave c = 5 with the working √25 = 5, angles 36.87° and 53.13° and area 6; legs 2 and 4 gave 2√5 ≈ 4.472135955; hypotenuse 13 with leg 12 gave a = 5; a hypotenuse shorter than the leg and three filled boxes each gave a message; 5, 12, 13 checked as a right triangle (25 + 144 = 169 = 13²), 5, 12, 14 as obtuse and 5, 12, 30 as no triangle; the page survived a reload and fitted a 390 px phone with no errors. We also checked the check: taking the square root of only the numerator of a fraction made 1,365 cases fail.
Sources
Checked on .
Frequently Asked Questions
Built by Derek Giordano · Part of Ultimate Design Tools