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Pythagorean theorem calculator

Enter two sides of a right triangle: the third shows, with the angles, area and perimeter. You can also check whether three lengths make a right angle.

Hypotenuse

5

Acute angles: 36.87° and 53.13°.
Area: 6. Perimeter: 12.

Quick challenge

Streak 0Best 0

Test yourself: every answer is computed by the tool, and the wrong choices are the most common traps.

How it works

Formula: c² = a² + b²

In a right triangle, the square of the hypotenuse (the side opposite the right angle, the longest) equals the sum of the squares of the other two sides: c² = a² + b². To find the hypotenuse, take the square root of a² + b²; to find a side, the square root of c² − a². The converse checks a right angle: if a² + b² = c², the triangle is a right triangle.

Worked examples

  • Sides of 3 and 4

    Hypotenuse: √(9 + 16) = 5.

  • A 16 ft ladder, its foot 4 ft from the wall

    It reaches 15.49 ft high: √(16² − 4²).

  • Diagonal of a 12 × 9 ft room

    15 ft.

Common mistakes

  • Adding the sides: 3 + 4 = 7.

    Add the squares, then take the square root: √(9 + 16) = 5.

  • Using the formula on a triangle without a right angle.

    The theorem only holds for right triangles: check it first with the tool.

  • Taking a side of the right angle for the hypotenuse.

    The hypotenuse is the longest side, opposite the right angle.

Common Pythagorean triples (whole-number sides)

Common Pythagorean triples (whole-number sides)
abc
345
51213
6810
72425
81517
91215
94041
121620
202129
123537

Frequently asked questions

What is the hypotenuse?

The side opposite the right angle, always the longest side of a right triangle.

How do I find a side if I know the hypotenuse?

b = √(c² − a²): with c = 13 and a = 5, b = √(169 − 25) = 12.

How do builders check a right angle?

With the 3-4-5 rule: 3 ft along one side, 4 ft along the other, and the diagonal should be 5 ft.

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