Free online Pythagorean theorem calculator: enter any two sides of a right triangle to get the hypotenuse or the missing leg, plus perimeter, area, both acute angles and radii
Two legs to hypotenuse, or hypotenuse plus one leg to the missing leg - every useful combination
Perimeter, area, both acute angles, circumradius and inradius are computed in the same pass
Enter three sides to test right-angle, obtuse and acute cases with the converse theorem
A to-scale sketch marks sides a, b and c plus the right-angle symbol for quick verification
Fill in 3-4-5, 5-12-13, 8-15-17 and more with one click, alongside a reference table
Everything runs in your browser - no uploads and no sign-up required
Choose "two legs → hypotenuse" when both legs are known, or the matching mode when you have the hypotenuse.
Type two positive values (decimals allowed) or click a common triple such as 3-4-5.
The result panel shows the solved side plus all three sides, perimeter, area, acute angles and radii; the diagram is drawn to scale.
Use the triple checker to test any three side lengths and see whether they form a right, acute or obtuse triangle.
Use the 3-4-5 triple to square corners on site, or calculate diagonal bracing lengths
Work out the length of a diagonal brace, stair stringer or angled cut from two known sides
Treat north-south and east-west offsets as the two legs to estimate point-to-point distance
Compute screen distances, collision radii and vector lengths with the same formula
Verify answers, demonstrate the formula rearrangements and explore triple patterns with the diagram
Find the diagonal from width and height, or check whether a set fits your space
In any right triangle the sum of the squares of the legs equals the square of the hypotenuse: a² + b² = c². The rule appears in ancient Chinese mathematics as the "3-4-5" relation in the Zhoubi Suanjing and is known in the West as the Pythagorean theorem. It is one of the most-proved theorems in mathematics, with demonstrations ranging from area rearrangements to similar triangles.
From a² + b² = c² you get c = √(a² + b²) for the hypotenuse, a = √(c² − b²) and b = √(c² − a²) for a missing leg. Always confirm that the hypotenuse is the longest side; otherwise the expression under the square root turns negative, which means the given sides cannot form a right triangle.
A Pythagorean triple is a set of positive integers satisfying a² + b² = c². The smallest is 3-4-5, and every integer multiple of a triple is again a triple (6-8-10, 9-12-15). Other common sets include 5-12-13, 8-15-17, 7-24-25 and 20-21-29. Builders use 3-4-5 or its multiples to check square corners quickly.
The converse states that if three sides satisfy a² + b² = c² (c the longest), the triangle is right-angled. If a² + b² is greater than c² the triangle is acute; if smaller it is obtuse. This makes it easy to classify a triangle from measured side lengths alone - useful in surveying and drafting.
| Triple (a-b-c) | Perimeter | Area | Acute angles ∠A / ∠B |
|---|---|---|---|
| 3-4-5 | 12 | 6 | 36.87° / 53.13° |
| 6-8-10 | 24 | 24 | 36.87° / 53.13° |
| 5-12-13 | 30 | 30 | 22.62° / 67.38° |
| 8-15-17 | 40 | 60 | 28.07° / 61.93° |
| 7-24-25 | 56 | 84 | 16.26° / 73.74° |
| 20-21-29 | 70 | 210 | 43.60° / 46.40° |