What is the Pythagorean theorem?
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². Here, c is the hypotenuse (the side opposite the 90° angle) and a and b are the two shorter sides called legs. The theorem works for any right triangle, regardless of its proportions.
How do you calculate the hypotenuse?
To find the hypotenuse when you know both legs, take the square root of the sum of their squares: c = √(a² + b²). For example, if a = 3 and b = 4, then c = √(9 + 16) = √25 = 5. You can also solve for a missing leg: if you know the hypotenuse c and one leg b, then a = √(c² − b²).
What is the Pythagorean theorem formula?
The formula is a² + b² = c², where a and b are the legs and c is the hypotenuse. Rearranging gives three solve forms:
- Find c (hypotenuse): c = √(a² + b²)
- Find a (leg): a = √(c² − b²)
- Find b (leg): b = √(c² − a²)
Each form requires c to be larger than either leg — the hypotenuse is always the longest side in a right triangle. For 3D geometry, the same principle extends to the cylinder calculator when computing diagonal lengths.
What are some Pythagorean theorem examples?
The most famous Pythagorean triple is 3-4-5: 3² + 4² = 9 + 16 = 25 = 5². Multiplying any triple by a constant gives another: 6-8-10, 9-12-15. Other common triples include 5-12-13 (25 + 144 = 169) and 8-15-17 (64 + 225 = 289). To find the missing leg when the hypotenuse is 13 and one leg is 5: a = √(169 − 25) = √144 = 12.
When is the Pythagorean theorem useful?
Construction workers use it to check that corners are square — a 3-4-5 or 5-12-13 layout confirms a 90° angle. Architects calculate diagonal bracing lengths. Navigators find straight-line distances between two points. Surveyors measure land boundaries. In everyday life, you can use it to find the diagonal of a screen or room. Use our tile calculator for related floor planning calculations.
How can you tell if a triangle is a right triangle?
The converse of the Pythagorean theorem states that if a² + b² = c² holds for three side lengths, the triangle must be right-angled. To check: square all three sides, add the two smaller results, and compare to the largest squared. Example — sides 5, 12, 13: 5² + 12² = 25 + 144 = 169 = 13². Right triangle ✓. Sides 5, 6, 8: 5² + 6² = 25 + 36 = 61 ≠ 64 = 8². Not a right triangle ✗. This is how builders verify corners in the field: a 3-4-5 layout always confirms 90°. For complete triangle analysis with angles and area, use our triangle calculator.
What are Pythagorean triples and why do they matter?
A Pythagorean triple is a set of three positive integers (a, b, c) satisfying a² + b² = c². The most useful triples in practice are 3-4-5, 5-12-13, 8-15-17, and 7-24-25. These produce exact integer results — no rounding needed. Builders and carpenters memorize common triples to quickly lay out right angles without a protractor. Any triple scaled by an integer is also a triple: 3-4-5 becomes 6-8-10, 12-16-20, and so on.