The Pythagorean Theorem Explained (With Examples)
The Pythagorean theorem says a² + b² = c² for any right triangle. Here's how to use it to find a missing side, test for right angles, and measure distance.
The Pythagorean theorem says that in any right triangle, a² + b² = c²: the square of the hypotenuse (c, the side opposite the right angle) equals the sum of the squares of the two shorter sides. That one relationship lets you find any missing side of a right triangle, test whether an angle is a right angle, and even measure distance on a grid.
This guide shows how to use it to find the hypotenuse, find a missing leg, test for a right angle with the converse, and calculate distance - each with a worked example - plus the triples worth memorizing.
The formula: a² + b² = c²
In a right triangle, the two sides that form the right angle are the legs (a and b), and the longest side, opposite the right angle, is the hypotenuse (c). The theorem ties them together:
Finding the hypotenuse
When you know both legs, square them, add, and take the square root. For legs of 3 and 4:
Finding a missing leg
If you know the hypotenuse and one leg, subtract instead of add. Say the hypotenuse is 13 and one leg is 5:
The converse: testing for a right angle
The theorem works in reverse too. If the three sides satisfy a² + b² = c², the triangle must have a right angle opposite the longest side. Check the sides 6, 8, and 10:
Pythagorean triples worth memorizing
A few whole-number right triangles come up again and again. Recognizing them lets you skip the arithmetic:
| Triple (a, b, c) | Why it works |
|---|---|
| 3, 4, 5 | 9 + 16 = 25 |
| 5, 12, 13 | 25 + 144 = 169 |
| 8, 15, 17 | 64 + 225 = 289 |
| 7, 24, 25 | 49 + 576 = 625 |
Using it to measure distance
The distance formula on a coordinate grid is just the Pythagorean theorem in disguise: the horizontal and vertical gaps between two points are the legs, and the straight-line distance is the hypotenuse.
For the points (1, 2) and (4, 6), the horizontal gap is 3 and the vertical gap is 4:
Common mistakes (and how to avoid them)
- Using the theorem on a triangle that isn't right-angled - it only works when there's a 90° angle.
- Mislabeling the hypotenuse - it's always the longest side, opposite the right angle.
- Adding when you should subtract - add to find the hypotenuse, subtract to find a leg.
- Forgetting the final square root and leaving your answer as c² instead of c.
- Dropping units - if the sides are in centimetres, the missing side is in centimetres too.
How a tutor makes geometry click
Right-triangle problems get easy once you can see which side is the hypotenuse and whether to add or subtract. A tutor working live on a shared whiteboard draws the figure, labels the sides, and coaches that decision until it's automatic - then connects it to area, trigonometry, and coordinate geometry. Our online geometry tutoring pairs you with a specialist who makes the theorem a tool you reach for with confidence.
Because the Pythagorean theorem underpins trigonometry, the distance formula, and much of coordinate geometry, getting it solid now pays off across the whole subject. Your first trial lesson is free.
The bottom line
The Pythagorean theorem, a² + b² = c², solves any right triangle. Add the squared legs and take the root to find the hypotenuse; subtract to find a leg; use the converse (a² + b² = c²) to test for a right angle; and apply it as the distance formula on a grid. Identify the hypotenuse, choose add or subtract, and the missing side follows every time.
Frequently asked questions
What is the Pythagorean theorem?+
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². It lets you find any one side of a right triangle when you know the other two.
How do you find the hypotenuse using the Pythagorean theorem?+
Square the two shorter sides (the legs), add them, and take the square root: c = √(a² + b²). For example, with legs of 3 and 4, c = √(9 + 16) = √25 = 5. The hypotenuse is always the longest side and sits opposite the right angle.
How do you find a missing leg (not the hypotenuse)?+
Rearrange the formula to subtract instead of add: a² = c² − b², so a = √(c² − b²). For instance, if the hypotenuse is 13 and one leg is 5, the other leg is √(169 − 25) = √144 = 12. The key is that you subtract when solving for a leg and add when solving for the hypotenuse.
Does the Pythagorean theorem work for all triangles?+
No - it only works for right triangles (triangles with a 90° angle). For triangles without a right angle, you use the law of sines or the law of cosines instead. In fact, you can use the theorem's converse to test whether a triangle is right-angled: if a² + b² = c², the angle opposite c is exactly 90°.
What are Pythagorean triples?+
Pythagorean triples are sets of three whole numbers that satisfy a² + b² = c², such as 3-4-5, 5-12-13, 8-15-17, and 7-24-25. Recognizing them lets you solve many right-triangle problems instantly, and any multiple of a triple (like 6-8-10) works too.
Can a tutor help me with the Pythagorean theorem and geometry?+
Yes - a tutor working live on a shared whiteboard can draw the triangle, show exactly which side is the hypotenuse, and coach the difference between finding a leg and finding the hypotenuse until it's automatic. That guided practice is what turns the formula into a tool you can use with confidence.
Maya Ellis is a mathematics specialist at iTutorzz. She focuses on making algebra, geometry, trigonometry, calculus, and statistics genuinely click for students, and writes iTutorzz's step-by-step math guides and worked examples.
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