Pythagorean Theorem Calculator
Pick the side you want to find and enter the other two. The calculator solves a² + b² = c² and also returns the triangle's area, perimeter and the two acute angles.
Result
- Sides a, b, c
- 3, 4, 5
- Area (½ × a × b)
- 6
- Perimeter (a + b + c)
- 12
- Angle α (opposite a)
- 36.8699°
- Angle β (opposite b)
- 53.1301°
The Pythagorean theorem
A right triangle has one 90° angle. The side opposite that angle is the hypotenuse, always the longest side, and the other two sides are the legs. The theorem states that the area of the square built on the hypotenuse equals the combined area of the squares on the two legs. In symbols:
a² + b² = c²
Rearranged for each unknown: c = √(a² + b²), a = √(c² − b²), b = √(c² − a²).
Worked example: legs of 3 and 4. 3² + 4² = 9 + 16 = 25, and √25 = 5, so the hypotenuse is 5. The area is ½ × 3 × 4 = 6, the perimeter is 3 + 4 + 5 = 12, and the acute angles are tan⁻¹(3/4) ≈ 36.87° and tan⁻¹(4/3) ≈ 53.13°, which add to 90° as they must.
How to use this calculator
- Choose which side is unknown: the hypotenuse or one of the legs.
- Enter the two known sides in any unit. The answer is in the same unit; the area is in that unit squared.
- If you entered a hypotenuse, it must be longer than the known leg, otherwise no right triangle exists.
Where it is used
- Construction: a 3-4-5 layout checks that a corner is square without a protractor.
- Distance: the straight-line distance between two points on a grid is √(Δx² + Δy²), the theorem in disguise.
- Screens and TVs: a diagonal size and aspect ratio give the width and height.
- Navigation and physics: combining perpendicular components of velocity or force.
For triangles of any shape, the area calculator handles base-and-height and three-side (Heron's formula) inputs.
Frequently asked questions
What is the Pythagorean theorem?
In any right triangle the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². It was known to Babylonian mathematicians and is named after the Greek philosopher Pythagoras.
How do I find a leg if I know the hypotenuse?
Rearrange the formula: a = √(c² − b²). For a hypotenuse of 13 and a leg of 12, a = √(169 − 144) = √25 = 5. The hypotenuse must be longer than the known leg or the subtraction goes negative.
Does the theorem work for triangles without a right angle?
No. For other triangles you need the law of cosines, c² = a² + b² − 2ab·cos(C), which reduces to the Pythagorean theorem when the angle C is 90° because cos 90° = 0.
What are Pythagorean triples?
Sets of three whole numbers that satisfy a² + b² = c², such as 3-4-5, 5-12-13, 8-15-17 and 7-24-25. Any multiple of a triple (6-8-10, 9-12-15) is also a triple, which builders use to lay out square corners with a tape measure.