Pythagorist
Geometry

Pythagoras Theorem Calculator

Find any side of a right-angled triangle using Pythagoras' theorem (a² + b² = c²). Enter any two sides to calculate the third, with full step-by-step working — from squaring each side to taking the square root.

Finding hypotenuse (c) — a² + b² = c²

Side c

5

How it works

  1. Write Pythagoras' theorem

    a2+b2=c2a^2 + b^2 = c^2

    In a right-angled triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides.

  2. Substitute known sides

    32+42=c23^2 + 4^2 = c^2

    Substitute a = 3, b = 4.

  3. Calculate squares

    9+16=c29 + 16 = c^2

    3² = 9, 4² = 16.

  4. Add

    c2=25c^2 = 25

    9 + 16 = 25.

  5. Square root

    c=25=5c = \sqrt{25} = 5

    c = √25 = 5.

    Result:c = 5

Pythagoras' theorem

a2+b2=c2a^2 + b^2 = c^2
aaOne shorter side (leg) of the right-angled triangle
bbThe other shorter side (leg)
ccThe hypotenuse — the longest side, opposite the right angle

Example substitution

32+42=9+16=25=52⇒c=53^2 + 4^2 = 9 + 16 = 25 = 5^2 \Rightarrow c = 5

The classic 3-4-5 right triangle. 9 + 16 = 25, so c = √25 = 5.

Worked examples

A right triangle has legs a = 3 and b = 4. Find the hypotenuse.

  1. 1
    Formula: a² + b² = c²
  2. 2
    Substitute: 3² + 4² = 9 + 16 = 25
  3. 3
    Square root: c = √25 = 5
Answer: c = 5

A right triangle has hypotenuse c = 13 and leg b = 5. Find a.

  1. 1
    Rearrange: a² = c² − b²
  2. 2
    Substitute: a² = 169 − 25 = 144
  3. 3
    Square root: a = √144 = 12
Answer: a = 12

A ladder 10 m long leans against a wall. The base is 6 m from the wall. How high does it reach?

  1. 1
    Setup: a² + 6² = 10²
  2. 2
    Calculate: a² = 100 − 36 = 64
  3. 3
    Square root: a = 8 m
Answer: 8 m

Frequently asked questions

What is the Pythagoras theorem?+

In a right-angled triangle, a² + b² = c², where c is the hypotenuse (longest side, opposite the right angle).

How do you find the hypotenuse?+

c = √(a² + b²). Square both shorter sides, add them, then take the square root.

How do you find a shorter side using Pythagoras?+

Rearrange: a = √(c² − b²). Subtract the square of the known short side from the hypotenuse squared.

What is a Pythagorean triple?+

Three whole numbers that satisfy a² + b² = c². Common examples: 3,4,5 and 5,12,13.

Who was Pythagoras?+

Pythagoras was an ancient Greek mathematician (c. 570–495 BC). The theorem bearing his name was known in Babylon centuries earlier, but he is credited with its proof.

How do I know which side is the hypotenuse?+

The hypotenuse is always the longest side and is always opposite the right angle (90° angle).

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