In any right triangle, the two shorter sides and the longest side obey one of the most famous relationships in mathematics: the square of the hypotenuse equals the sum of the squares of the legs. It’s called the Pythagorean theorem, it’s about 2,500 years old, and it still solves a huge share of geometry problems — from rooflines to GPS distances. Here’s how to use it flawlessly.
The Theorem
For a right triangle with legs a and b and hypotenuse c (the side across from the 90° angle):
a^2 + b^2 = c^2
Three things follow immediately:
- Finding the hypotenuse: c = √(a² + b²) — add the squares, then root.
- Finding a leg: a = √(c² − b²) — subtract the known leg’s square from the hypotenuse’s square, then root.
- Checking for a right angle: if a² + b² = c² holds exactly, the triangle is right-angled (the converse of the theorem).
The hypotenuse is always the longest side and always sits across from the right angle. Before writing a single number, point at it and label it c — most errors start with c in the wrong place.
Worked Example 1: Finding the Hypotenuse
A right triangle has legs of 9 cm and 12 cm. Find the hypotenuse.
- Write the formula: a² + b² = c²
- Substitute: (9)² + (12)² = c²
- Square: 81 + 144 = c² → c² = 225
- Take the root: c = √225 = 15 cm
This is the 3-4-5 triple scaled by 3 — which is exactly why memorizing triples (below) pays off.
Worked Example 2: Finding a Missing Leg
Find leg a in a right triangle where leg b = 5 and hypotenuse c = 13.
- Write the formula: a² + b² = c²
- Substitute: a² + (5)² = (13)²
- Square: a² + 25 = 169
- Isolate a² (subtract!): a² = 169 − 25 = 144
- Take the root: a = √144 = 12
The missing leg is 12 — the 5-12-13 triple. Notice the subtraction: when solving for a leg, you always take the hypotenuse’s square minus the known leg’s square.
Worked Example 3: Real-World Application
A 15-foot ladder leans against a wall with its base 9 feet from the wall. How high up the wall does it reach?
- Identify c: the ladder itself is the hypotenuse, c = 15. The ground distance is a leg, b = 9. The wall height is the unknown leg, a.
- Set up: a² + 9² = 15²
- Compute: a² + 81 = 225 → a² = 144
- Root: a = 12 feet
Common Pythagorean Triples to Memorize
Triples are integer sets that satisfy the theorem exactly. Spotting one saves the entire calculation:
- 3 – 4 – 5 (and multiples: 6-8-10, 9-12-15, 30-40-50)
- 5 – 12 – 13 (and multiples: 10-24-26)
- 8 – 15 – 17
- 7 – 24 – 25
If you see two numbers from a triple, the third is free. On a timed test, that’s pure profit.
3 Mistakes to Avoid
- Adding squares when solving for a leg: c² − b² = a², not c² + b². Adding gives an answer bigger than the hypotenuse — impossible, since the hypotenuse is the longest side.
- Misidentifying the hypotenuse: putting a leg’s value into the c position corrupts everything downstream. The hypotenuse is across from the right angle, always.
- Forgetting the final square root: a² + b² gives you c², not c. Stopping at 225 instead of 15 is the most painful way to lose a mark.
Sanity check every answer: each leg must be shorter than the hypotenuse. If your “leg” comes out longer than c, you’ve swapped the subtraction — fix it before moving on.