The Pythagorean theorem in simple words
A right triangle has one square corner, like the corner of a page. The two sides that meet at that corner are the legs, a and b. The longest side, across from the corner, is the hypotenuse, c.
The Pythagorean theorem says a² + b² = c². In words: square each leg (multiply it by itself), add the two answers, and you get the square of the hypotenuse. Take the square root to find c.
Example: legs of 3 and 4. 3 × 3 = 9 and 4 × 4 = 16. 9 + 16 = 25, and the square root of 25 is 5, so the hypotenuse is 5.
To find a missing leg, do it the other way round: square the hypotenuse, take away the square of the leg you know, then take the square root.
Formula and method
For a right triangle, a²+b²=c², where c is the hypotenuse opposite the 90° angle. To find c, use √(a²+b²). To find a leg, subtract the other leg’s square from c² before taking the square root.
Worked example
With a = 3 and b = 4, c = √(9+16) = 5. The area is 3 × 4 / 2 = 6 square units and perimeter is 12 units. A = atan(3/4) ≈ 36.86989765° and B ≈ 53.13010235°.
Inputs and limitations
This theorem applies only to right triangles. For a general triangle use the Triangle Calculator. The hypotenuse must be longer than either leg. Convert all side measurements to the same unit first; the area is in that unit squared.
Whole-number right triangles
Each row satisfies a²+b²=c². Multiplying all three sides by the same positive factor preserves the right angle.
| Leg a | Leg b | Hypotenuse c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 20 | 21 | 29 |
Exact answers and worked steps
The steps substitute your numbers: c = √(3² + 4²) = √(9 + 16) = √25 = 5, or b = √(13² − 5²) = √(169 − 25) = √144 = 12. If whole-number sides do not give a perfect square, the answer is also shown in simplified radical form. Legs 1 and 1 give c = √2, and legs 2 and 4 give c = √20 = 2√5.
The results also include the inradius, (a + b − c) ÷ 2, and the circumradius, c ÷ 2.