📐 Pythagorean Theorem

The rule of right triangles

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What is the Pythagorean Theorem?

The Pythagorean theorem is a rule about right triangles — triangles that have one square corner (a 90° angle). It connects the lengths of the three sides.

  • The two shorter sides that form the right angle are called the legs (\(a\) and \(b\)).
  • The longest side, opposite the right angle, is the hypotenuse (\(c\)).
\[ a^2 + b^2 = c^2 \]

A classic 3-4-5 right triangle: \(3^2 + 4^2 = 9 + 16 = 25 = 5^2\)

Find the hypotenuse

\(c = \sqrt{a^2 + b^2}\)

Use this when you know both legs.

Find a leg

\(a = \sqrt{c^2 - b^2}\)

Use this when you know the hypotenuse and one leg.

Right angle?

If \(a^2 + b^2 = c^2\)

then the triangle has a right angle. If not, it doesn't!

Common triples

3, 4, 5   ·   5, 12, 13 8, 15, 17   ·   7, 24, 25

Whole-number side lengths that always work.

Worked Example

A ladder leans against a wall. Its base is 6 m from the wall and it reaches 8 m up. How long is the ladder?

1. The ladder is the hypotenuse \(c\); the legs are \(a = 6\) and \(b = 8\).
2. \(c^2 = a^2 + b^2 = 6^2 + 8^2 = 36 + 64 = 100\)
3. \(c = \sqrt{100} = 10\)
4. The ladder is 10 m long.

Tips

  • The theorem only works for right triangles.
  • The hypotenuse \(c\) is always the longest side — across from the right angle.
  • To find a leg, subtract inside the square root; to find the hypotenuse, add.
  • Don't forget the final square root!