Method · Geometry
How to Use the Pythagorean Theorem
Three steps, and the only decision is whether the side you want is the hypotenuse or one of the legs.
Identify the hypotenuse first
Knowing how to use the Pythagorean theorem starts with one identification, and getting it wrong makes everything afterwards meaningless.
c is always the hypotenuse: the side opposite the right angle, and always the longest. a and b are the two legs, and they are interchangeable — the theorem does not care which is which.
Finding the hypotenuse
Both legs known, hypotenuse wanted. Square, add, take the root.
- a² + b² = c²the theorem
- 5² + 12² = c²substitute
- 25 + 144 = 169square each and add
- c = 13take the positive root
The last step matters. Every positive number has two square roots, but a side length is a distance, so only the positive one is meaningful.
Finding a missing leg
The hypotenuse and one leg known. Rearrange first: subtract rather than add.
- a² = c² − b²rearrange for the leg
- a² = 100 − 36substitute
- a² = 64evaluate
- a = 8take the positive root
Adding here instead of subtracting is the single most common error in this topic, and it gives an answer larger than the hypotenuse — which is impossible and therefore easy to catch.
When the answer is not a whole number
Most of the time it is not, and the exact form is usually what is wanted.
- 9 + 25 = 34add the squares
- c = √34the exact answer
- ≈ 5.83the decimal, rounded at the end
√34 does not simplify, since 34 has no square factor. When it does — √50, say — simplify it: 5√2 is the finished form.
Using the converse to test for a right angle
The theorem works in reverse. If the three sides satisfy a² + b² = c², the triangle must be right-angled.
| Sides | a² + b² vs c² | The angle opposite c |
|---|---|---|
| 3, 4, 5 | 25 = 25 | exactly 90° |
| 4, 5, 6 | 41 < 36? no — 41 > 36 | acute |
| 3, 4, 6 | 25 < 36 | obtuse |
Builders use this with a tape measure: mark 3 units along one edge and 4 along the other, and the corner is square when the diagonal reads exactly 5.
Recognising the triples saves time
A handful of right triangles have whole-number sides, and spotting one removes the arithmetic entirely.
| Triple | Check | Multiples that also work |
|---|---|---|
| 3, 4, 5 | 9 + 16 = 25 | 6-8-10, 9-12-15, 15-20-25 |
| 5, 12, 13 | 25 + 144 = 169 | 10-24-26, 15-36-39 |
| 8, 15, 17 | 64 + 225 = 289 | 16-30-34 |
Any multiple of a triple is a triple, because scaling every side by the same factor scales each square by the factor squared and the equation survives. Seeing 6 and 8 and recognising 10 is faster than squaring anything.
Word problems: find the right triangle first
Most applications hide the triangle, and drawing it is most of the work.
- the ladder is the hypotenuseit is opposite the right angle at the ground
- h² = 25 − 9 = 16rearrange for the missing leg
- h = 4 mhow far up the wall it reaches
The same shape appears in diagonals of rectangles, distances on a grid, and the height of anything measured from a distance. Sketch it, label the right angle, and the theorem does the rest.
Questions about how to use the pythagorean theorem
How do I know which side is c?
It is the side opposite the right angle, and always the longest. If your c is not the longest side, the triangle is mislabelled.
How do I find a leg rather than the hypotenuse?
Rearrange to a² = c² − b² and subtract. An answer larger than the hypotenuse means you added instead.
What if the answer is not a whole number?
That is normal. Leave it as an exact square root, simplified if possible, and give the decimal only if asked.
Can I use it to check a corner is square?
Yes — that is the converse. If a² + b² equals c² for the three measured sides, the angle opposite c is exactly 90°.