Math Tools
Pythagorean Theorem Calculator
Find the hypotenuse or a missing leg of a right triangle using the Pythagorean theorem.
Pythagorean Theorem Calculator
Find the hypotenuse or a missing side of a right triangle using the Pythagorean theorem.
Use a² + b² = c² to find the hypotenuse.
Formula
c = √(a² + b²)
Hypotenuse (c)
5
Recommended Measuring Tool
Working with real right-triangle measurements?
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How it works
The Pythagorean theorem applies to right triangles and states that a² + b² = c², where c is the hypotenuse. This calculator can find either the hypotenuse or a missing leg.
What is the Pythagorean theorem?
The Pythagorean theorem is one of the most important formulas in geometry. It applies only to right triangles and shows the relationship between the two shorter sides and the longest side.
If the legs are called a and b, and the hypotenuse is called c, then the theorem states that a² + b² = c². This lets you solve for the hypotenuse or find a missing side when the other two sides are known.
Pythagorean theorem formula
Find the hypotenuse
a² + b² = c²
c = √(a² + b²)
Find a missing leg
c² - a² = b²
b = √(c² - a²)
Examples for finding the hypotenuse
These quick examples show how the Pythagorean theorem calculator works for the selected mode.
| Known sides | Result |
|---|---|
| a = 3, b = 4 | 5 |
| a = 5, b = 12 | 13 |
| a = 8, b = 15 | 17 |
Common right triangle triples
These common integer side combinations are useful for geometry, construction, and checking right triangles quickly.
| Triangle sides | Notes |
|---|---|
| 3, 4, 5 | Most common beginner example |
| 5, 12, 13 | Common geometry example |
| 8, 15, 17 | Useful for larger right triangles |
| 7, 24, 25 | Classic integer triple |
| 9, 40, 41 | Common reference triple |
How to use the Pythagorean theorem step by step
First, make sure the triangle is a right triangle. Then identify which sides are known. If you know both legs, use the formula c = √(a² + b²) to find the hypotenuse.
If you know the hypotenuse and one leg, use b = √(c² - a²) to find the missing leg. The calculator does this automatically and gives you the result instantly.
When a Pythagorean theorem calculator is useful
A Pythagorean theorem calculator is useful in geometry homework, trigonometry, construction, carpentry, surveying, architecture, and engineering. It helps whenever you need to find the diagonal, hypotenuse, or an unknown side in a right triangle.
It is also useful for practical real-life tasks such as checking wall braces, roof slopes, ladders against walls, rectangular room diagonals, and layout measurements on building projects.
Pythagorean theorem FAQ and common right triangle questions
What is the formula for the Pythagorean theorem?
The formula is a² + b² = c², where a and b are the legs of a right triangle and c is the hypotenuse.
How do you find the hypotenuse of a right triangle?
Square both legs, add the results, and take the square root of the total. This gives the hypotenuse.
How do you find a missing side of a right triangle?
If you know the hypotenuse and one leg, subtract the square of the known leg from the square of the hypotenuse, then take the square root.
Can the Pythagorean theorem be used for any triangle?
No. It only works for right triangles, which have one 90-degree angle.
What is the hypotenuse in a right triangle?
The hypotenuse is the longest side of a right triangle and it is always opposite the right angle.
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