In a right triangle, the two shorter sides (the legs) and the longest side (the hypotenuse) are linked by one equation:
a² + b² = c²
The hypotenuse c is always the side across from the right angle. It works for every right triangle and for no other kind.
Finding the hypotenuse
Square both legs, add, then take the square root.
Example The legs are 6 and 8. Find the hypotenuse.
- 6² + 8² = 36 + 64 = 100.
- c = √100 = 10.
Example The legs are 5 and 7. Find the hypotenuse.
- 5² + 7² = 25 + 49 = 74.
- c = √74 ≈ 8.60. Leave it as √74 if the question asks for an exact answer.
Finding a leg
Subtract the known leg’s square from the hypotenuse’s square.
Example The hypotenuse is 13 and one leg is 5. Find the other leg.
- 5² + b² = 13², so 25 + b² = 169.
- b² = 144, so b = 12.
Put the hypotenuse on its own
c must be the longest side, the one across from the right angle. If you add the hypotenuse to a leg instead of subtracting, your “leg” comes out longer than the hypotenuse, which is impossible. Use that as a check.
Pythagorean triples
Some right triangles have whole-number sides. Recognizing them saves time on tests.
| Triple | Multiples you’ll see |
|---|---|
| 3, 4, 5 | 6, 8, 10 · 9, 12, 15 · 15, 20, 25 |
| 5, 12, 13 | 10, 24, 26 |
| 8, 15, 17 | 16, 30, 34 |
| 7, 24, 25 | 14, 48, 50 |
If a triangle’s legs are 30 and 40, you can see it is 3-4-5 scaled by 10, so the hypotenuse is 50, with no squaring needed.
Is it a right triangle?
The theorem also works in reverse. Put the longest side in as c and compare.
- a² + b² = c²: right triangle.
- a² + b² > c²: acute triangle.
- a² + b² < c²: obtuse triangle.
Example Is a triangle with sides 9, 12 and 15 a right triangle?
81 + 144 = 225 = 15². Yes.
Example What kind of triangle has sides 7, 8 and 11?
49 + 64 = 113, and 11² = 121. Since 113 < 121, it is obtuse.
Real problems
Example A 13-foot ladder leans against a wall with its base 5 feet from the wall. How high does it reach?
The ladder is the hypotenuse. 5² + h² = 13², so h² = 144 and h = 12 feet.
Example A TV screen is 48 inches wide and 27 inches tall. What size is it?
Screens are sold by their diagonal. √(48² + 27²) = √(2304 + 729) = √3033 ≈ 55.1 inches, so it’s a 55-inch TV.
The distance formula
The distance between two points on a graph is the hypotenuse of a right triangle whose legs are the horizontal and vertical gaps. Written as a formula:
d = √((x₂ − x₁)² + (y₂ − y₁)²)
Example Find the distance from (1, 2) to (7, 10)
- The horizontal gap is 6 and the vertical gap is 8.
- d = √(36 + 64) = √100 = 10.
Example Find the distance from (−3, 4) to (2, −8)
d = √((2 − (−3))² + (−8 − 4)²) = √(25 + 144) = √169 = 13.
In three dimensions
The longest diagonal of a box with length l, width w and height h is √(l² + w² + h²). A box that is 3 by 4 by 12 has a diagonal of √(9 + 16 + 144) = √169 = 13.
Special right triangles
Two right triangles come up so often that their side ratios are worth memorizing.
| Triangle | Side ratio | If the shortest side is 1 |
|---|---|---|
| 45°-45°-90° | x : x : x√2 | 1, 1, √2 |
| 30°-60°-90° | x : x√3 : 2x | 1, √3, 2 |
In a 30-60-90 triangle, the hypotenuse is twice the shortest side, and the shortest side is across from the 30° angle. Both ratios come straight from the Pythagorean theorem: 1² + 1² = (√2)², and 1² + (√3)² = 2².
Try a few
The legs of a right triangle are 9 and 12. Find the hypotenuse.
Show answer
15. This is the 3-4-5 triple times 3.
The hypotenuse is 25 and one leg is 7. Find the other leg.
Show answer
24. 625 − 49 = 576, and √576 = 24.
Is a triangle with sides 10, 24 and 26 a right triangle?
Show answer
Yes. 100 + 576 = 676 = 26².
Find the distance between (−1, −2) and (5, 6).
Show answer
10. The gaps are 6 and 8.
A square has a diagonal of 10. How long is each side?
Show answer
5√2 ≈ 7.07. A square’s diagonal makes a 45-45-90 triangle, so the side is 10/√2.
You walk 12 blocks east and 5 blocks north. How far are you from where you started, in a straight line?
Show answer
13 blocks. This is a 5-12-13 triangle.
A 30-60-90 triangle has a hypotenuse of 14. Find the other two sides.
Show answer
7 and 7√3 ≈ 12.12. The short side is half the hypotenuse.
Practice
Questions students ask
Does it work for triangles that aren’t right?
No. For other triangles, use the law of cosines, c² = a² + b² − 2ab cos C. When C = 90°, cos C = 0 and it becomes the Pythagorean theorem.
Is it given on the SAT?
Yes. The SAT reference sheet includes a² + b² = c² and both special right triangles. You still save time by knowing them.
What should I learn next?
Right-triangle trigonometry, which finds sides and angles from a single side and an angle. See SOH CAH TOA.