Pythagorean Theorem Calculator
- Know both legs: a = 3 ft and b = 4 ft.
- c² = a² + b² = 3² + 4² = 9 + 16 = 25.
- c = √25 = 5 ft.
- Check: a² + b² = 9 + 16 = 25, and c² = 25.
- Height to the hypotenuse: 2.4 ft
- Inradius (inscribed circle): 1 ft
- Circumradius (half the hypotenuse): 2.5 ft
- Pythagorean triple: (3, 4, 5), a primitive triple
- a² + b² = 25 and c² = 25: they match, so it is a right triangle.
- Angles: 36.8699° + 53.1301° + 90° = 180°.
Pythagorean Theorem Calculator: Find the Missing Side of a Right Triangle
This free Pythagorean theorem calculator finds the hypotenuse or either leg of a right triangle when you know the other two sides. Choose the side you want to find, enter the two lengths you know, and the answer appears with each step of the working. It also gives the area, perimeter, both acute angles and a diagram that redraws as you change the numbers.
Quick answer: a right triangle with legs of 3 and 4 has a hypotenuse of 5, because 3² + 4² = 9 + 16 = 25 and the square root of 25 is 5.
The Pythagorean Theorem Formula
In a right triangle the two shorter sides, called legs, are a and b, and the longest side, opposite the right angle, is the hypotenuse c. The theorem states a² + b² = c². Rearranged, the hypotenuse is c = √(a² + b²), and a missing leg is a = √(c² − b²) or b = √(c² − a²). Since a leg can never be as long as the hypotenuse, the calculator warns you if the numbers you enter cannot form a right triangle.
Finding the Hypotenuse
Square each leg, add the results, and take the square root. For legs of 6 and 8, the hypotenuse is √(36 + 64) = 10. The answer is not always a whole number: two legs of 1 give √2, which is about 1.4142. This is the same calculation used to find the diagonal of a rectangle or the straight-line distance between two points on a grid.
Finding a Missing Leg
When the hypotenuse is known, subtract instead. With a hypotenuse of 13 and a leg of 5, the other leg is √(169 − 25) = √144 = 12. Choose Side a or Side b at the top of the calculator, enter the two sides you have, and the working is shown one step at a time.
Pythagorean Triples
A Pythagorean triple is a set of three whole numbers that satisfy the theorem. They are handy for builders and teachers because the sides come out as clean numbers. Any multiple of a triple is also a triple.
| Triple | a² + b² | c² | Multiples |
|---|---|---|---|
| 3, 4, 5 | 9 + 16 | 25 | 6-8-10, 9-12-15 |
| 5, 12, 13 | 25 + 144 | 169 | 10-24-26 |
| 8, 15, 17 | 64 + 225 | 289 | 16-30-34 |
| 7, 24, 25 | 49 + 576 | 625 | 14-48-50 |
| 20, 21, 29 | 400 + 441 | 841 | 40-42-58 |
| 9, 40, 41 | 81 + 1,600 | 1,681 | 18-80-82 |
Where the Theorem Is Used
Carpenters use the 3-4-5 rule to check that a corner is square. Ladder safety depends on it, since the reach of a ladder is a leg of a right triangle. It gives the length of a roof rafter, the diagonal of a screen and the shortest path across a field. Once you have a triangle's sides, the area calculator can work with its area, and the circle calculator covers round shapes. The scientific calculator is useful for roots and trigonometry, and the quadratic calculator helps when a geometry problem turns into an equation.
How to Use This Pythagorean Theorem Calculator
Choose the side to find
Pick Hypotenuse c, Side a or Side b.
Enter the two known sides
Type the lengths or drag the sliders, and pick a unit.
Read the answer
See the missing side, with each step of the working shown.
Check the extras
Get the area, perimeter, both angles, the height to the hypotenuse and a diagram.