The Pythagorean theorem
The idea
Right triangles obey a remarkable bargain: square the two legs, add those squares, and you get exactly the square of the hypotenuse — a² + b² = c², where c is the longest side, the one facing the right angle. You can literally picture it with the areas you already know: build a square on each side of the triangle, and the two smaller squares' areas together equal the largest one's. It is the standard tool for any straight-line distance you cannot measure directly.
Two cautions keep the theorem honest. First, it works only for right triangles — apply it to a triangle without a 90° angle and the equation simply lies to you. Second, resist the shortcut a + b = c; sides do not add, their squares do, which is why a diagonal shortcut is shorter than walking the two legs but still longer than either leg alone. Always identify which side is the hypotenuse before substituting, since c must be the side opposite the right angle.
Worked example
A rectangular field is 60 m by 80 m. Instead of walking along two sides from one corner to the opposite corner, you cut straight across the diagonal. How long is the shortcut, and how much walking does it save?
- The two sides and the diagonal form a right triangle, with the diagonal as the hypotenuse because it faces the field's 90° corner.
- Apply the theorem: diagonal² = 60² + 80² = 3600 + 6400 = 10000.
- Take the square root to recover the length: √10000 = 100 m for the diagonal shortcut.
- Compare with the two-side route: 60 + 80 = 140 m, so the shortcut saves 140 − 100 = 40 m.
- Sanity-check the geometry: 100 m is longer than either single side but shorter than the two sides combined, exactly what a straight-line shortcut should be.
Answer. The diagonal shortcut is 100 m long and saves 40 m of walking.
Check your understanding
- Why must the theorem use the squares of the side lengths rather than the lengths themselves?
- How can you use the theorem in reverse to test whether a triangle contains a right angle?
- What happens to the diagonal if both sides of the field are doubled, and why?
- Where would the calculation go wrong if you mistakenly treated one of the legs as the hypotenuse?
Build the foundations first
The Pythagorean theorem builds on these concepts. If any feel shaky, start there.