Hypotenuse Calculator — Pythagorean Theorem
Find the hypotenuse c from both legs a and b — or find a missing leg given the other two sides. The calculator also gives both acute angles, area, and perimeter of the right triangle.
Solve for
a² + b² = c²
- 1
Square leg a
a² = 3² = 9 - 2
Square leg b
b² = 4² = 16 - 3
Sum of squares
a² + b² = 9 + 16 = 25 - 4
Hypotenuse c = √(a² + b²)
√25 = 5
How does this calculator work?
c = √(a² + b²) for a right triangle; rearrange to find a leg: a = √(c² − b²). Solves for any one side given the other two. Also gives acute angles A = arctan(a/b), B = arctan(b/a), area = ½ab, and perimeter = a + b + c.
Formula
How this is calculated
The Pythagorean theorem states that in any right triangle the square of the hypotenuse (the side opposite the 90° angle) equals the sum of the squares of the two legs: a² + b² = c². Rearranging gives the missing side directly — no trigonometry needed when you know two sides.
The two acute angles follow from the inverse tangent: angle A = arctan(a/b) and angle B = arctan(b/a). Since the angles of any triangle sum to 180° and one is 90°, A + B = 90°. Area is ½ × a × b (the two perpendicular legs serve as base and height), and perimeter is a + b + c.
The theorem applies to any right triangle in flat (Euclidean) geometry, regardless of units or size. For non-right triangles, use the law of cosines: c² = a² + b² − 2ab cos C.
Frequently asked questions
Use c = √(a² + b²). For the classic 3-4-5 triangle: c = √(9 + 16) = √25 = 5. Enter a = 3 and b = 4 in this calculator and it returns c = 5 exactly.
Integer solutions such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25) are Pythagorean triples — a² + b² = c² with whole numbers. Any positive-integer multiple of a triple is also a triple.
No — the Pythagorean theorem and these rearrangements apply only when one angle is exactly 90°. For arbitrary triangles, use the law of cosines: c² = a² + b² − 2ab cos C, which reduces to the Pythagorean theorem when C = 90°.
TG we-Calculate Editorial Team. (2026). Hypotenuse Calculator — Pythagorean Theorem [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/hypotenuse-calculator
TG we-Calculate Editorial Team. "Hypotenuse Calculator — Pythagorean Theorem." TG we-Calculate. 2026. https://we-calculate.com/calculator/hypotenuse-calculator.
TG we-Calculate Editorial Team, "Hypotenuse Calculator — Pythagorean Theorem," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/hypotenuse-calculator
@misc{wecalculate_hypotenuse_calculator, title = {Hypotenuse Calculator — Pythagorean Theorem}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/hypotenuse-calculator}}, year = {2026}, note = {TG we-Calculate} }
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