Pythagorean Theorem Calculator
Solve a right triangle from any two sides using a² + b² = c². Get the missing side, area, perimeter and angles instantly, with the check shown.
About the Pythagorean Theorem Calculator
The Pythagorean theorem says that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². This calculator solves for whichever side you are missing — enter the two legs to find the hypotenuse (c = √(a² + b²)), or one leg and the hypotenuse to find the other leg (b = √(c² − a²)).
Beyond the missing side, it also gives the triangle's area (½ × a × b), perimeter (a + b + c) and both acute angles, and it displays the a² + b² = c² check so you can verify the arithmetic. It validates your input too: the hypotenuse must always be the longest side, so entering a leg longer than c is flagged instead of producing a wrong answer.
Students meet this theorem in every geometry course, but it is just as useful off the page: carpenters use the 3-4-5 rule to square corners, and it computes ladder reach, TV screen dimensions, diagonal distances across a room or plot, and the straight-line distance between two points.
How to Use the Pythagorean Theorem Calculator
- 1Choose what to solve for: the hypotenuse c, or one of the legs.
- 2Enter the two sides you know.
- 3Read the missing side, area, perimeter and angles.
- 4Check the a² + b² = c² line to see the verification.
Frequently Asked Questions
How do I find the hypotenuse of a right triangle?
Square both legs, add them, and take the square root: c = √(a² + b²). With legs 3 and 4: c = √(9 + 16) = √25 = 5. This 3-4-5 triangle is the classic example — any multiple (6-8-10, 9-12-15) is also a right triangle.
How do I find a leg when I know the hypotenuse?
Rearrange the formula: b = √(c² − a²). If the hypotenuse is 13 and one leg is 5, the other leg is √(169 − 25) = √144 = 12. Select the leg option in the calculator and it applies this automatically.
Why does the calculator say my hypotenuse is invalid?
The hypotenuse is always the longest side of a right triangle — it faces the 90° angle. If you enter a hypotenuse smaller than or equal to a leg, c² − a² would be zero or negative, and no real triangle exists. Swap the values: the largest side must go in the hypotenuse field.
Does the Pythagorean theorem work for any triangle?
No, only right triangles. 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°. In fact, checking whether a² + b² equals c² is a standard test for whether a triangle is right-angled.
How is the Pythagorean theorem used in real life?
Builders check square corners with the 3-4-5 rule; a 10 ft ladder with its base 6 ft from a wall reaches √(100 − 36) = 8 ft up; a '55 inch' TV is its diagonal, so a 16:9 screen is about 48 × 27 inches; and navigation apps compute straight-line distances with the same formula.