Enter any two sides of a right triangle — legs a and b, or hypotenuse c — and this calculator solves a² + b² = c² for the missing length. Optional √ fields let you type 2√5 the way the theorem is written. Results include both acute angles, area, perimeter, the altitude to the hypotenuse, and worked steps.
Worked identities
| Given | Missing side |
|---|---|
| a = 3, b = 4 | c = 5 |
| b = 4, c = 5 | a = 3 |
| a = 1, b = 1 | c = √2 ≈ 1.4142135623731 |
| a = 2, b = 2 | c = 2√2 ≈ 2.8284271247462 |
| a = 5, b = 12 | c = 13 |
Examples
Classic 3-4-5 triangle
a = 3, b = 4. c = √(9 + 16) = √25 = 5. ∠α = arcsin(0.6) = 36.87° = 36°52'12" = 0.6435 rad. Area = 6, perimeter = 12, h = 12 / 5 = 2.4.
Coefficient times a square root (default)
a = 2 and c = 2√5. b = √((2√5)² − 2²) = √(20 − 4) = 4. The same triangle is a 1-2-√5 right triangle scaled by 2.
Isosceles right triangle
a = 1, b = 1. c = √2 ≈ 1.4142135623731. Both acute angles are 45° = π/4, and h = √2 / 2 ≈ 0.70710678118655.
Formula
In a right triangle the square of the hypotenuse equals the sum of the squares of the legs:
a² + b² = c²
Solve for whichever side is unknown:
c = √(a² + b²)
a = √(c² − b²)
b = √(c² − a²)
The acute angles, area, perimeter, and altitude to the hypotenuse follow once all three sides are known:
∠α = arcsin(a / c)
∠β = arcsin(b / c)
area = a × b / 2
perimeter = a + b + c
h = a × b / c
The default example is a = 2 and c = 2√5. Then b = √(20 − 4) = √16 = 4, area = 4, and h = 4√5 / 5 ≈ 1.7888543819998.
Triangle limits
The relationship a² + b² = c² applies only to a right triangle, with c opposite the 90° angle. Measurements that do not satisfy the relationship may reflect rounding or a triangle that is not right. Derived angles and lengths inherit input uncertainty and are rounded for display.