About the Pythagorean Theorem Calculator
Pythagorean Theorem Calculator finds the missing side of a right triangle when you know the other two. Type any two of the sides a, b and c, where c is the hypotenuse, and leave the third box empty. The tool shows the answer, the working as a formula with your numbers, and extra facts about the triangle: its area, perimeter, both acute angles, the height to the hypotenuse and the radius of its inscribed circle. If your sides are whole numbers that satisfy a² + b² = c², such as 3, 4 and 5, it tells you that it is a Pythagorean triple. It also warns you when the numbers cannot make a right triangle, for example when the hypotenuse is not the longest side. It runs in your browser and works with decimals.
Updated September 2026
How to use the Pythagorean Theorem Calculator
- 1Decide which side you are looking for: a leg (a or b) or the hypotenuse (c).
- 2Type the two sides you know into their boxes and leave the third empty.
- 3Read the missing side in the big result box.
- 4Check the working line to see the formula with your numbers.
- 5Use the smaller boxes for the area, perimeter and angles.
Where the theorem is used
The theorem is one of the most used results in geometry. It gives the diagonal of a rectangle or a screen, the length of a ladder needed to reach a window, the straight-line distance between two points on a map grid, and the length of a rafter for a roof of a known height and span.
A 55-inch TV, for example, is measured along the diagonal. If it is 48 inches wide, the height follows from the theorem: √(55² − 48²) is about 26.9 inches. Carpenters use a 3-4-5 triangle scaled up, such as 6, 8 and 10 feet, to check that a corner is square.
Step by step
- Identify the right angle and the hypotenuse, which is opposite to it.
- If you need the hypotenuse, add the squares of the legs and take the square root.
- If you need a leg, subtract the square of the known leg from the square of the hypotenuse, then take the square root.
- Check that the hypotenuse is the longest side.
- Round only at the end, so small errors do not build up.
Angles and other results
Once all three sides are known, the acute angles follow from the ratios of the sides. The angle opposite side a is arctan(a ÷ b). The two acute angles always add up to 90 degrees. The area is half the product of the legs, and the height to the hypotenuse is the product of the legs divided by the hypotenuse.
The inradius, the radius of the largest circle that fits inside the triangle, equals (a + b − c) ÷ 2, and the circumscribed circle has a radius of c ÷ 2, because the hypotenuse is its diameter.
Tips and related tools
If your answer looks strange, check that you did not put a leg in the hypotenuse box. For triangles with no right angle, or when you know angles rather than two sides, use Triangle Calculator. For the size of shapes, Area Calculator and Volume Calculator help, and Exponent, Root and Log Calculator does the squares and square roots.
Angle Converter turns the angles into radians if you need them for a formula, and Quadratic Equation Solver may come in when you need to solve for a side from an equation.
Features
- Finds any one missing side from two
- Shows the working with your numbers
- Area, perimeter and both acute angles
- Height to the hypotenuse and inradius
- Spots whole-number Pythagorean triples
- Clear messages when the sides cannot form a right triangle
- Works with decimals, and runs in your browser
Frequently asked questions
What is the Pythagorean theorem?
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². The hypotenuse is the longest side, opposite the right angle. If the legs are 3 and 4, then c² = 9 + 16 = 25, so c = 5.
How do I find the hypotenuse?
Square both legs, add the squares and take the square root: c = √(a² + b²). Enter the two legs in this calculator, leave c empty and the tool does that and shows the working.
How do I find a missing leg?
Subtract the square of the known leg from the square of the hypotenuse, then take the square root: a = √(c² − b²). The hypotenuse must be longer than the leg you know, otherwise no right triangle exists, and the calculator tells you so.
What is a Pythagorean triple?
It is a set of three whole numbers that satisfy a² + b² = c², such as 3-4-5, 5-12-13, 8-15-17 and 7-24-25. Multiples of a triple, such as 6-8-10, also work. Builders use the 3-4-5 rule to check a square corner.
Does the theorem work for triangles that are not right triangles?
No. It only applies when one angle is exactly 90 degrees. For other triangles you need the law of cosines, which reduces to the Pythagorean theorem when the angle is 90 degrees. Triangle Calculator solves triangles in general.
Can I use any units?
Yes, as long as both sides use the same unit. The answer is in that same unit, and the area is in square units. Convert first with Unit Converter if your measurements are in different units.