📐 Right Triangle & Hypotenuse Solver
Pythagorean Theorem Calculator
Solve $a^2 + b^2 = c^2$ for hypotenuse or leg lengths with step-by-step exact radical simplification and 3D distance solving.
Pythagorean Triples:
3 - 4 - 5 Triple
5 - 12 - 13 Triple
8 - 15 - 17 Triple
7 - 24 - 25 Triple
Hypotenuse Length (c)
c = 5
Triangle Area
6.00 sq units
Perimeter (a+b+c)
12.00 units
Angle α (opp a)
36.87° (0.64 rad)
Angle β (opp b)
53.13° (0.93 rad)
Mathematical Formulas & Radical Simplification
The Pythagorean Theorem relates the three sides of any Euclidean right triangle:
c = √(a² + b²)
a = √(c² − b²)
3D Distance = √(Δx² + Δy² + Δz²)
a = √(c² − b²)
3D Distance = √(Δx² + Δy² + Δz²)
To express solutions in exact radical form, factor out the largest perfect square $k^2$ dividing the radicand: $\sqrt{N} = k\sqrt{M}$ (e.g. $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$).