The Pythagorean Theorem states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the two legs: a² + b² = c². If you know any two sides of a right triangle, you can find the third by rearranging this equation and taking a square root.
Example: a right triangle with legs of 3 and 4 has a hypotenuse of 5, because 3² + 4² = 9 + 16 = 25, and √25 = 5.
Worked Example 1: Finding the Hypotenuse
Problem: A right triangle has legs of length 6 and 8. Find the length of the hypotenuse c.
- Write the theorem: a² + b² = c².
- Substitute the leg values: 6² + 8² = c².
- Square each term: 36 + 64 = c².
- Add: 100 = c².
- Take the positive square root: c = √100 = 10.
The hypotenuse is 10 units. (Notice this is a 6-8-10 triangle — a multiple of the classic 3-4-5 Pythagorean triple.)
The Method in 4 Steps
- Identify the hypotenuse. It is always the side opposite the right angle and always the longest side. Example: in a triangle with sides 5, 12, 13, the hypotenuse is 13.
- Write a² + b² = c², with c as the hypotenuse and a, b as the legs. Example: 5² + 12² = 13².
- Substitute the two known values and simplify the squares. Example: 25 + 144 = c², so c² = 169.
- Solve for the missing side. If you're finding the hypotenuse, take the square root of the sum. If you're finding a leg, subtract first, then take the square root. Example: c = √169 = 13.
Finding a Leg Instead of the Hypotenuse
If the hypotenuse is known, rearrange to a² = c² − b² before taking the square root. The largest side always sits alone on the right of a² + b² = c².
Worked Example 2: Finding a Missing Leg
Problem: A right triangle has a hypotenuse of 17 and one leg of 8. Find the other leg b.
- Write the theorem with the hypotenuse as c: a² + b² = c².
- Substitute the known values: 8² + b² = 17².
- Square: 64 + b² = 289.
- Isolate b²: b² = 289 − 64 = 225.
- Take the positive square root: b = √225 = 15.
The missing leg is 15 units. (This is an 8-15-17 Pythagorean triple.)
Common Mistakes to Avoid
- Treating a leg as the hypotenuse. The hypotenuse is always the longest side and always opposite the right angle. Plugging the hypotenuse in for a or b gives a wrong (often negative under the root) answer.
- Adding when you should subtract. When solving for a leg, you must compute c² − (known leg)², not add the two given values.
- Forgetting to take the square root. Students often stop at c² = 100 and write c = 100. The final step is √.
- Using it on non-right triangles. a² + b² = c² only holds for right triangles. For other triangles, use the Law of Cosines.
- Squaring the sum instead of each side. (a + b)² ≠ a² + b². Square each leg separately, then add.
You now have the formula and the steps — but the fastest way to make it stick is to work a few problems where a tutor asks the right question at the right moment instead of just handing you the answer. That's what LernOS does. Try a guided Pythagorean Theorem problem below and see the difference.