The Pythagorean theorem
In a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
Example: legs of 3 and 4 give c = √(9 + 16) = √25 = 5. The angles are 36.87° and 53.13°, the area is 6 and the perimeter 12. With a hypotenuse of 13 and a leg of 5, the other leg is √(169 − 25) = 12.
Common Pythagorean triples
These whole-number right triangles come up constantly: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29, and any multiple of them, like 6-8-10.
Builders use 3-4-5 to check a corner is square: measure 3 ft along one wall and 4 ft along the other; if the diagonal is exactly 5 ft, the corner is 90°. For roof angles and rise over run, see the slope calculator.
Frequently asked questions
How do you find the hypotenuse?
Square both legs, add them, and take the square root: c = √(a² + b²).
Does the Pythagorean theorem work for all triangles?
No, only right triangles. For other triangles, use the law of cosines: c² = a² + b² − 2ab × cos(C).
What is the 3-4-5 rule?
A triangle with sides 3, 4 and 5 is always a right triangle, so builders use it to check that corners are square.
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