Pythagoras Triangle Calculator — Solve Any Right Triangle
Enter any two sides of a right triangle and instantly find the third side, both acute angles, the area, perimeter, and the altitude from the right angle to the hypotenuse.
Pythagorean triple — all three sides are whole numbers!
- 1
Square leg a
3² = 9 - 2
Square leg b
4² = 16 - 3
Sum a² + b²
9 + 16 = 25 - 4
Hypotenuse c = √(a² + b²)
√25 = 5The square root of the sum of squares gives the length of the hypotenuse.
How does this calculator work?
Enter any two sides of a right triangle. The calculator applies a² + b² = c² to find the missing side, then derives both acute angles (A = arcsin(a/c), B = 90°−A), the area (a·b/2), the perimeter (a+b+c), and the altitude to the hypotenuse (a·b/c). Pythagorean triples (integer sides) are flagged automatically.
Formula
How this is calculated
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse c equals the sum of the squares of the two legs a and b: a² + b² = c². The hypotenuse is always the side opposite the right (90°) angle and is always the longest side. Knowing any two sides determines the third uniquely.
Given two legs, c = √(a² + b²). Given the hypotenuse and one leg, the missing leg = √(c² − known_leg²). The calculator also detects whether all three sides are whole numbers — a Pythagorean triple such as (3, 4, 5) or (5, 12, 13). Both acute angles follow from the inverse sine: angle A = arcsin(a/c) and angle B = 90° − A.
The altitude drawn from the right-angle vertex perpendicular to the hypotenuse has length h_c = a·b / c. This altitude also creates two smaller right triangles that are similar to the original — a relationship used in geometric proofs and in computing the projections a·a/c and b·b/c of each leg onto the hypotenuse. All inputs must be in the same unit; outputs use the same unit for lengths and square units for area.
Frequently asked questions
Square both legs, add them, then take the square root: c = √(a² + b²). For example, with a = 3 and b = 4, c = √(9 + 16) = √25 = 5.
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c², such as (3, 4, 5), (5, 12, 13), and (8, 15, 17). These represent right triangles with all whole-number sides.
It is the perpendicular distance from the right-angle vertex to the hypotenuse. Its length is h = a·b / c, and it is geometrically important because it divides the right triangle into two smaller triangles that are both similar to the original.
Also known as
TG we-Calculate Editorial Team. (2026). Pythagoras Triangle Calculator — Solve Any Right Triangle [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/pythagoras-triangle-calculator
TG we-Calculate Editorial Team. "Pythagoras Triangle Calculator — Solve Any Right Triangle." TG we-Calculate. 2026. https://we-calculate.com/calculator/pythagoras-triangle-calculator.
TG we-Calculate Editorial Team, "Pythagoras Triangle Calculator — Solve Any Right Triangle," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/pythagoras-triangle-calculator
@misc{wecalculate_pythagoras_triangle_calculator, title = {Pythagoras Triangle Calculator — Solve Any Right Triangle}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/pythagoras-triangle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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