Pythagorean Theorem Calculator
The missing side of a right triangle, with its angles, area and perimeter.
Good to know. Builders check a square corner with 3, 4 and 5: if the diagonal is 5, the corner is right.
How it works
In plain words
In a right-angled triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the other two. So the hypotenuse is the square root of a² + b², and a shorter side is the square root of c² − a².
The formula
c = √(a² + b²) b = √(c² − a²) angle opposite a = arctan(a ÷ b)
Worked example
With sides 3 and 4: 3² + 4² = 9 + 16 = 25, and √25 = 5. The hypotenuse is 5.
Sources
No outside data or rates: this is plain arithmetic.
3 automated tests check its answers on every change.
About the pythagorean theorem calculator
Pythagoras’ theorem says that in any triangle with a right angle, the square of the longest side equals the sum of the squares of the two shorter sides: a² + b² = c². The longest side, opposite the right angle, is the hypotenuse.
Choose whether you know the two shorter sides and want the hypotenuse, or know the hypotenuse and one side and want the other. The calculator also gives both acute angles, which always add up to 90°, and the area and perimeter. Any unit works, as long as all sides use the same one.
It comes up far beyond school maths. The diagonal of a TV or phone screen is the hypotenuse of its width and height. A ladder 5 m long, with its foot 1.5 m from a wall, reaches √(25 − 2.25) ≈ 4.77 m up. Builders square a corner with the 3-4-5 rule: mark 3 m along one wall and 4 m along the other, and if the diagonal between the marks is exactly 5 m, the corner is a right angle.
Sets of whole numbers that fit, like 3, 4, 5 and 5, 12, 13 and 8, 15, 17, are called Pythagorean triples.
Questions people ask
How do I find the hypotenuse?
Square the two shorter sides, add them, and take the square root. For 6 and 8: 36 + 64 = 100, and √100 = 10.
How do I find a missing shorter side?
Square the hypotenuse, subtract the square of the known side, and take the square root. For c = 13 and a = 5: √(169 − 25) = 12.
Does the Pythagorean theorem work for every triangle?
Only for triangles with a right angle. Other triangles need the law of cosines.