What is the Pythagorean theorem?
A right triangle has one 90° corner. The side opposite it is the hypotenuse, always the longest side; the other two are the legs. The Pythagorean theorem says that in any right triangle, a² + b² = c², so if you know two sides you can always work out the third.
When the answer isn’t a whole number, it is a square root such as √13. We show it simplified (√52 is 2√13), and as a decimal to 4 places. The other two angles come from the legs: angle A is tan⁻¹(a ÷ b), and the two always add up to 90°.
Source OpenStax · Prealgebra 2e, 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem ↗Common Core State Standards · Grade 8, apply the Pythagorean Theorem (8.G.B.7) ↗OpenStax · Precalculus 2e, 6.3 Inverse Trigonometric Functions ↗
How to use the Pythagorean theorem
Here it is with your numbers:
- c² = 3² + 4²
- c² = 9 + 16
- c² = 25
- c = √25
- c = 5
It means the side opposite the right angle is 5 long, in the same unit as the legs.
Pythagorean triple examples
A Pythagorean triple is three whole numbers that fit a² + b² = c², so the triangle has no square roots at all. Multiply a triple by any whole number and you get another one: 3, 4, 5 becomes 6, 8, 10. Tap a row to try it.
FAQs
What is the Pythagorean theorem in simple terms?
In any right triangle, the two shorter sides squared add up to the longest side squared: a² + b² = c². The longest side, c, is the hypotenuse, opposite the right angle; a and b are the legs.
Source OpenStax · Prealgebra 2e, 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem ↗
How to find the 3rd side length of a triangle?
For a right triangle, square the two sides you know. If they are the legs, add the squares and take the square root: legs 3 and 4 give √25, which is 5. If you know the hypotenuse and a leg, subtract instead: 13² − 5² is 144, so the other leg is 12.
Why are 3, 4, 5, and 5, 12, 13 pythagorean triples?
Because all three are whole numbers that fit a² + b² = c²: 3² + 4² is 9 + 16, which is 25, or 5²; and 5² + 12² is 25 + 144, which is 169, or 13². Any multiple of a triple works too, such as 6, 8, 10.
Sources
- OpenStax: Prealgebra 2e, 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
- Common Core State Standards: Grade 8, apply the Pythagorean Theorem (8.G.B.7)
- OpenStax: Precalculus 2e, 6.3 Inverse Trigonometric Functions