The Pythagorean Theorem
One of the most useful formulas ever discovered. If you know two sides of a right triangle, you can always find the third.
~8 min3 lessonsInteractive triangle5-question quiz
01
a² + b² = c²
It only works for right triangles: triangles with one 90° corner. The two sides that make the corner are a and b. The longest side, opposite the corner, is the hypotenuse, c.
a² + b² = c²the squares of the two short sides add up to the square of the long side
Example · the classic 3-4-5 triangle
- 3² + 4² = 9 + 16 = 25
- c² = 25, so c = √25
- c = 5
02
Finding any missing side
Missing the long side (c)? Square the two short sides, add, then square root.
Example · a = 6, b = 8
- 6² + 8² = 36 + 64 = 100
- c = √100
- c = 10
Missing a short side? Square the long side, subtract the square of the other short side, then square root.
Example · c = 13, a = 5
- 13² − 5² = 169 − 25 = 144
- b = √144
- b = 12
Sanity check: the hypotenuse is always the longest side. If your answer for c is shorter than a or b, something went wrong.
03
Where you'll actually use it
- TV and phone sizes are measured diagonally. A 16:9 TV with a 55″ diagonal is about 48″ wide and 27″ tall.
- Ladders: a 5 m ladder with its foot 1.5 m from the wall reaches √(25 − 2.25) ≈ 4.77 m up.
- Squaring a corner: builders mark 3 units along one side and 4 along the other. If the diagonal is exactly 5, the corner is a perfect right angle.
- Shortcut distances: walking diagonally across a 30 m × 40 m field is 50 m instead of 70 m around the edge.
Try it · triangle calculator