Pythagorean Triangle Calculator

Enter any two sides of a right triangle to find the third side, all angles, area, and perimeter.

Enter Any Two Sides

Leave the unknown side empty. Side c is the hypotenuse (longest side).

Result

Missing Side
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units

Step-by-Step Solution

a² + b² = c²

Understanding the Pythagorean Triangle

A Pythagorean triangle (or right triangle) is a triangle with one angle equal to 90 degrees. The side opposite the right angle is called the hypotenuse (c), and the other two sides are called legs (a and b). The Pythagorean theorem states that a² + b² = c².

Key Formulas

Find Hypotenuse

When you know both legs a and b.

c = sqrt(a² + b²)

Find a Leg

When you know the hypotenuse and one leg.

a = sqrt(c² - b²)

Area of Right Triangle

Half the product of the two legs.

Area = (1/2) x a x b

Angles in a Right Triangle

In a right triangle, one angle is always 90 degrees. The other two acute angles can be found using trigonometric functions. If you know sides a (opposite) and c (hypotenuse), then angle A = arcsin(a/c). The two acute angles always sum to 90 degrees.

Common Pythagorean Triples

  • (3, 4, 5) -- The most basic Pythagorean triple: 9 + 16 = 25.
  • (5, 12, 13) -- Another common primitive triple: 25 + 144 = 169.
  • (8, 15, 17) -- 64 + 225 = 289.
  • (7, 24, 25) -- 49 + 576 = 625.

Verification

You can always verify your answer by checking that a² + b² = c². If the values satisfy this equation, the triangle is a valid right triangle. This calculator automatically verifies the result for you.