Key takeaways
- The Pythagorean theorem is c² = a² + b², for right triangles only.
- a and b are the two legs; c is the hypotenuse, the longest side.
- Take the square root at the end to get the actual length, not the squared value.
- 3-4-5 and 5-12-13 are classic whole-number examples, called Pythagorean triples.
In this guide
What is the Pythagorean theorem?
In any right triangle (a triangle with one 90° corner), the two shorter sides are called legs, and the longest side, opposite the right angle, is called the hypotenuse. The Pythagorean theorem connects the three.
Why does the Pythagorean theorem matter?
It's one of the most useful formulas in everyday geometry: it tells you the length of a side you can't directly measure, as long as you know the other two.
A ladder resting 3 m from a wall and reaching 4 m up needs to be exactly 5 m long: √(3² + 4²) = √25 = 5. That's the famous 3-4-5 triangle, and it's a genuine right angle every time.The formula
| Symbol | What it means | Example |
|---|---|---|
| a | One leg of the right triangle | 3 m |
| b | The other leg | 4 m |
| c | The hypotenuse: the longest side, opposite the right angle | 5 m |
How to find the hypotenuse step by step
- 1Identify the two legs. The two shorter sides that meet at the right angle.
- 2Square each leg. Multiply each one by itself.
- 3Add the two squared values together.
- 4Take the square root. This gives you the hypotenuse, c.
Now try it yourself. Change the two legs in the calculator below and watch the hypotenuse update, with every step shown.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Solving for the wrong side. If you already know the hypotenuse and one leg, rearrange to
b² = c² - a², not add. - Skipping the square root.
c²is the squared hypotenuse; take the square root to get the actual length c. - Mixing up which side is the hypotenuse. It's always the longest side, directly opposite the right angle.
Tips and tricks
- To check if a corner is truly square on site, measure 3 units along one wall and 4 along the other; the diagonal between those points should be exactly 5 units.
- The same idea, applied to coordinates on a graph, becomes the distance formula.
Where you'll use it in real life
- Construction: checking that a corner or foundation is a true right angle.
- DIY: finding the right ladder length to reach a given height from a given distance.
- Navigation: finding the straight-line distance between two points on a grid map.
- Design: verifying screen or TV diagonal sizes from width and height.
Quick summary
To find the hypotenuse of a right triangle, square both legs, add them together, then take the square root. The theorem only applies to right triangles. Use the calculator above to check your own triangle with every step shown.
Worked example
A ladder's base sits 3 meters from a wall, and it needs to reach a point 4 meters up the wall. How long does the ladder need to be?
- 01c² = a² + b²
- 02c² = 3² + 4²
- 03c² = 9 + 16
- 04c² = 25
- 05c = 5
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do I find a missing leg instead of the hypotenuse?
Rearrange the formula to a² = c² - b², then take the square root. Subtract the known leg squared from the hypotenuse squared before taking the root.
Does the Pythagorean theorem work on any triangle?
No, only on right triangles, ones with exactly one 90° angle. For other triangles, use the [law of cosines](/calculator/law-of-cosines) instead.
What's a common example of Pythagorean triple numbers?
3-4-5 and its multiples, like 6-8-10 or 9-12-15, are classic examples where all three sides are whole numbers.
How is the Pythagorean theorem related to the distance formula?
The [distance formula](/calculator/distance-formula) for two points on a graph is really the Pythagorean theorem applied to the horizontal and vertical gaps between the points.
