Pythagoras Theorem Calculator
Find the missing side, area, and perimeter of a right-angled triangle using the Pythagorean Theorem.
Find Hypotenuse (c)
Right Triangles
What Is the Pythagorean Theorem?
The Pythagorean theorem is a geometry rule for right triangles. It says the square of the hypotenuse equals the sum of the squares of the two legs.
Core rule
a and b are the legs that meet at the right angle. c is the hypotenuse, the longest side.
Calculator Uses
What This Calculator Can Find
Use the calculator when you know two sides of a right triangle and need the third side, area, or perimeter.
- Find the hypotenuse
- Enter the two legs of a right triangle to calculate the longest side.
- Find a missing leg
- Enter the hypotenuse and one known leg to solve for the other leg.
- Area and perimeter
- After all three sides are known, the calculator also finds triangle area and perimeter.
- Step-by-step math
- Review the square, subtraction or addition, square root, and final side length.
The Math
Pythagorean Theorem Formulas
The same relationship can be rearranged depending on which side is missing.
| Need to find | Formula | When to use it |
|---|---|---|
| Pythagorean theorem | a and b are legs. c is the hypotenuse. | |
| Find hypotenuse | Use when both legs are known. | |
| Find leg a | Use when c and b are known. | |
| Find leg b | Use when c and a are known. | |
| Triangle area | Right triangle area uses the two perpendicular legs. | |
| Perimeter | Add all three side lengths. |
How to Use
How to Use This Pythagorean Theorem Calculator
Choose the missing side, enter the two known values, calculate, and review the right-triangle measurements.
- 1
Choose the unknown side
Select hypotenuse c, missing leg a, or missing leg b.
- 2
Enter two known sides
Use positive values and make sure the hypotenuse is the longest side.
- 3
Calculate
Click the calculate button to find the missing side.
- 4
Review area and perimeter
After all sides are known, check the extra triangle measurements and steps.
Examples
Common Pythagorean Triples
Some right triangles have whole-number side lengths. These are useful for quick checks and mental math.
| Leg a | Leg b | Hypotenuse c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 6 | 8 | 10 |
| 7 | 24 | 25 |
| 8 | 15 | 17 |
| 9 | 12 | 15 |
| 20 | 21 | 29 |
Real Life
Where the Pythagorean Theorem Is Used
The theorem is useful whenever two perpendicular distances form a diagonal.
| Use case | Why it matters |
|---|---|
| Construction | Check whether a corner is square and estimate diagonal braces. |
| Navigation | Find straight-line distance when horizontal and vertical distances are known. |
| Screens and rectangles | Find a diagonal from width and height. |
| Coordinate geometry | Use the distance formula between two points. |
| School geometry | Solve right triangle side-length problems with steps. |
Common Mistakes
Pythagorean Theorem Mistakes to Avoid
The biggest issues are using the theorem on non-right triangles or treating a leg as the hypotenuse.
- Using it on any triangle
- The Pythagorean theorem works only for right triangles with a 90-degree angle.
- Confusing the hypotenuse
- The hypotenuse is always the longest side and sits opposite the right angle.
- Forgetting the square root
- After calculating c^2, you still need the square root to get c.
- Entering an impossible hypotenuse
- The hypotenuse must be longer than either leg. A hypotenuse smaller than a leg cannot form a right triangle.
Frequently Asked Questions
It finds the missing side of a right triangle using the formula a^2 + b^2 = c^2.
The formula is a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse.
Use c = square root of a^2 + b^2.
Use a = square root of c^2 - b^2 or b = square root of c^2 - a^2.
The hypotenuse is the longest side of a right triangle. It is always opposite the 90-degree angle.
No. It works only for right triangles.
A Pythagorean triple is a group of three whole numbers that satisfy a^2 + b^2 = c^2, such as 3, 4, 5 or 5, 12, 13.
Square the two shorter sides and add them together. If the result equals the square of the longest side, the triangle is a right triangle.
Yes. After finding all three sides, it calculates area as a times b divided by 2 and perimeter as a + b + c.
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