Quick answer: With legs 3 and 4 the hypotenuse is √(3² + 4²) = √25 = 5; with hypotenuse 13 and leg 5 the other leg is √(13² − 5²) = √144 = 12.
How it's calculated
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.
So c = √(a² + b²), and a leg is √(c² − b²). The hypotenuse is always the longest side, so a leg can only be found when c is longer than the known leg.
Whole-number solutions such as 3-4-5, 5-12-13 and 8-15-17 are called Pythagorean triples; builders use 3-4-5 to check a right angle. For angles and the rest of the triangle, use the right triangle calculator.
Worked examples
- Legs 3 and 4: c = √(9 + 16) = 5
- Hypotenuse 13, leg 5: b = √(169 − 25) = 12
- Legs 1 and 1: c = √2 = 1.41421356
Frequently asked questions
Does the theorem work for any triangle?
No, only for right triangles. For other triangles use the law of cosines in the triangle calculator.
Which side is the hypotenuse?
The side opposite the 90° angle. It is always the longest side.
How do I check that a corner is square?
Measure 3 units along one side and 4 along the other: if the diagonal is 5, the corner is 90°. Multiples such as 60, 80 and 100 cm work the same way.
What's the hypotenuse if the other sides are 3 and 4?
5. Nine plus sixteen is 25, and the square root of 25 is 5.
How long a ladder do I need to reach 4 m up a wall?
About 4.12 m, if you set the foot 1 m out from the wall, which is the usual safe angle. Add a bit more if you need to step off at the top.
How do I find the diagonal of a rectangle?
Treat the two sides as the legs. A 3 × 4 m room is 5 m corner to corner, and a TV screen 121 cm wide and 68 cm tall is about 139 cm, or 55 inches.