A right triangle has legs a = 3 and b = 4. Find the hypotenuse c.

Step-by-step solution

1. 1

   Identify which side is the hypotenuse. In a right triangle the hypotenuse is the side opposite the right angle, and it is always the longest side. The two given sides, $3$ and $4$, are the legs — the sides that form the right angle.

2. 2

   State the Pythagorean theorem. For legs $a,b$ and hypotenuse $c$:

   $a^2+b^2=c^2$

   The theorem only applies with the hypotenuse alone on one side; that is why identifying it first matters.

3. 3

   Substitute the leg lengths.

   $3^2+4^2=c^2$

4. 4

   Add the squares.

   $9+16=25\quad\Longrightarrow\quad c^2=25$

5. 5

   Take the positive square root. A length cannot be negative, so only the positive root is kept:

   $c=\sqrt{25}=5$

6. 6

   Note why the answer is so clean. $(3,4,5)$ is a Pythagorean triple — all three sides are whole numbers. Most leg pairs do not do this: change the legs to $6$ and $5$ and the hypotenuse becomes the irrational $\sqrt{61}$.