Math Calculator
Pythagorean Theorem Calculator
Find the missing side of a right triangle using a² + b² = c². Solve for the hypotenuse, solve for a missing leg, check a triangle, and view area, perimeter, angles, altitude, and a simple visual chart.
For a right triangle, c is the hypotenuse and a, b are the two legs.
Calculate a right triangle
Choose a mode, enter the known sides, and get the missing side with step-by-step work and triangle measurements.
Right triangle visual
This right triangle has legs 3 and 4, so the hypotenuse is 5.
Step-by-step calculation
c = √(3² + 4²) = √(9 + 16) = √25 = 5
How to use the Pythagorean theorem
The Pythagorean theorem works only for right triangles. It connects the two legs and the hypotenuse using the equation a² + b² = c². The hypotenuse is the longest side and is opposite the 90-degree angle.
Find hypotenuse: c = √(a² + b²)
Find leg a: a = √(c² - b²)
Find leg b: b = √(c² - a²)
If your answer includes decimals, CalcMora’s decimal to fraction calculator can help convert the result into a fraction. For number-system practice, you can also use the decimal to binary converter.
Common Pythagorean theorem examples
| Known legs a, b | Hypotenuse c | Note |
|---|---|---|
| 3, 4 | 5 | Classic 3-4-5 triangle |
| 5, 12 | 13 | Common Pythagorean triple |
| 8, 15 | 17 | Integer right triangle |
| 7, 24 | 25 | Large right triangle |
| 9, 40 | 41 | Useful for geometry practice |
What else does this calculator show?
Area
The area of a right triangle is a × b ÷ 2, where a and b are the legs.
Perimeter
The perimeter is the total distance around the triangle: a + b + c.
Angles
The two smaller angles are calculated from the side lengths using trigonometry.
Where the Pythagorean theorem is useful
This theorem is used in geometry, construction, design, navigation, computer graphics, surveying, diagonal measurements, ladder problems, map distance, and many school math problems. For example, if a wall height and ground distance are known, the theorem can estimate the ladder length needed to reach the top.
Frequently asked questions
What is the Pythagorean theorem?
The Pythagorean theorem states that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse: a² + b² = c².
How do I find the hypotenuse?
If you know legs a and b, use c = √(a² + b²). For example, if a = 3 and b = 4, then c = √(9 + 16) = 5.
How do I find a missing leg?
If you know the hypotenuse and one leg, subtract the known leg squared from the hypotenuse squared, then take the square root. For example, a = √(c² - b²).
Can this calculator check if three sides make a right triangle?
Yes. Enter all three sides in Check Triangle mode. The calculator checks whether a² + b² is approximately equal to c².
What is the hypotenuse?
The hypotenuse is the longest side of a right triangle. It is always opposite the 90-degree angle.