Find the hypotenuse or a missing leg of a right triangle using a² + b² = c².
Named after the ancient Greek mathematician Pythagoras, this theorem relates the three sides of any right triangle: the square of the hypotenuse (c, the side opposite the right angle) equals the sum of the squares of the other two sides (the legs, a and b).
Given both legs, square each one, add them together, then take the square root.
Given the hypotenuse and one leg, square both, subtract the leg's square from the hypotenuse's square, then take the square root.
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 6 | 8 | 10 |
For legs a=6 and b=8: c = √(6² + 8²) = √(36+64) = √100 = 10.
Builders and carpenters use it to check that a corner is truly square, since a 3-4-5 triangle only closes correctly at a right angle. It's also how GPS and mapping software estimate straight-line ("as the crow flies") distance between two points, how TV and monitor sizes are measured diagonally, and how engineers calculate the shortest support brace across a rectangular frame.
What is the Pythagorean theorem? For any right triangle, a² + b² = c², where c is the hypotenuse (the side opposite the 90° angle) and a, b are the two legs.
Does this theorem work for any triangle? No — it only applies to right triangles, meaning one angle must be exactly 90°. For other triangles, use the Law of Cosines instead.
Can I use this to check if a triangle is a right triangle? Yes — if a² + b² equals c² for the three side lengths (with c the longest side), the triangle is a right triangle.
What are common Pythagorean triples? Whole-number solutions like 3-4-5, 5-12-13, 8-15-17, and 7-24-25 — and any multiple of them, such as 6-8-10.
Who discovered the Pythagorean theorem? It's named for the Greek mathematician Pythagoras (c. 570–495 BCE), though evidence shows Babylonian and other mathematicians knew and used the relationship before him.
How is the Pythagorean theorem used in everyday life? Beyond geometry class, it shows up in construction (squaring corners), navigation and mapping (straight-line distance), screen sizing (diagonal measurements), and any situation where you know two perpendicular distances and need the direct distance between their endpoints.
What is the Pythagorean theorem equation? a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle) and a and b are the two legs. Enter any two of the three values above to solve for the third.
A 10 ft ladder has its base 2.5 ft from a wall. The height it reaches is √(10² − 2.5²) = √93.75 ≈ 9.68 ft. Safety guidance commonly recommends a base distance of about one quarter of the working height of the ladder, so right triangles are a practical way to check setups.
A 55-inch television is measured along the diagonal. With a 16:9 shape, its width is about 47.9 inches and its height about 27.0 inches, because the diagonal of a 16 by 9 rectangle is √(16² + 9²) ≈ 18.36 units long and the screen is scaled up to match 55 inches.
Builders use the 3-4-5 triangle to check that a corner is exactly 90 degrees. Measure 3 units along one wall, 4 units along the other, and the distance between those marks must be 5 units. Any multiple works too, such as 6-8-10 or 9-12-15, which is convenient for large layouts like decks and foundations.
Can the sides be decimals? Yes. The theorem works for any positive lengths, and the results are often irrational numbers that you round to a practical precision.
Which side is the hypotenuse? It is the longest side, opposite the right angle. Always put it as c.