Pythagorean theorem
Also known as: a² + b² = c², pythagoras' theorem
The Pythagorean theorem states that in a right triangle, the squares of the two legs add up to the square of the hypotenuse: a² + b² = c². It only applies to right triangles.
The Pythagorean theorem describes the relationship among the three sides of a right triangle: if the legs (the two sides forming the 90° angle) have lengths a and b, and the hypotenuse (the side opposite the right angle) has length c, then a² + b² = c². Given any two sides of a right triangle, the theorem lets you solve for the third.
For example, a triangle with legs of 5 and 12 has a hypotenuse of 13, because 5² + 12² = 25 + 144 = 169 = 13². Whole-number side sets like this are called Pythagorean triples — 3-4-5, 5-12-13, 8-15-17, and their multiples (6-8-10, 10-24-26) appear constantly on standardized tests. Recognizing a triple saves you from computing square roots under time pressure.
The theorem works only on right triangles — that restriction is itself a test point. Its converse is also useful: if a triangle's sides satisfy a² + b² = c², the triangle must be right. When a² + b² is greater than c², the triangle is acute; when it is less, the triangle is obtuse. The theorem also underlies the distance formula in coordinate geometry and the side ratios of the special 45-45-90 and 30-60-90 triangles.
The Pythagorean theorem is among the most heavily tested geometry facts on the SAT, GRE, and CLT. Expect it inside word problems — ladders against walls, diagonal distances, TV screen sizes — and layered into right triangle trigonometry questions where you find a missing side before computing a ratio.
Key takeaways
- In a right triangle, a² + b² = c², where c is the hypotenuse.
- The theorem applies only to right triangles — never to acute or obtuse triangles.
- Memorize common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, and their multiples.
- The converse identifies right triangles: sides satisfying a² + b² = c² guarantee a 90° angle.
- The distance formula and special right triangles (45-45-90, 30-60-90) are direct applications.