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.
This on the SAT
The Digital SAT tests the Pythagorean Theorem in its Geometry and Trigonometry domain, both directly and hidden inside other question types. Direct wording: In right triangle ABC, angle B is a right angle, AB = 6, and BC = 8. What is the length of AC? Hidden inside the distance formula: What is the distance between (1, 2) and (5, 5) in the xy-plane? — that distance is √(4² + 3²) = 5, the hypotenuse of a 3-4-5 right triangle. Hidden inside a diagonal problem: the diagonal of a rectangle with sides 9 and 12 is the hypotenuse of a right triangle with legs 9 and 12, which is 15 (a scaled 3-4-5).
Memorizing the common Pythagorean triples pays off across every one of these SAT variants: 3-4-5, 5-12-13, 8-15-17, 7-24-25, plus any integer multiple (6-8-10, 9-12-15, 10-24-26, and so on). Bluebook (the digital testing app) ships with Desmos and lets you use it on every question, so you can plot two points and read the distance directly on coordinate-plane versions — but on the more common label-a-triangle format, spotting the triple is faster than typing anything.
Prepping for the SAT? Take our free SAT math diagnostic — a short diagnostic that tells you which Digital SAT topics to focus on, including the Geometry and Trigonometry questions built around right triangles, trig ratios, and the distance formula. No card, no email up front.
For the sin/cos/tan side of right-triangle problems, see our SAT-scoped guide: SAT right triangle trigonometry: SOHCAHTOA. For the pure hypotenuse-finding walkthrough with more worked examples, see How to Find the Hypotenuse of a Right Triangle.
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.