Pythagorean Theorem Examples: Solved Problems

In this article you'll find a collection of solved examples, from the simplest to the more challenging. Each one is solved step by step.

Calculating the hypotenuse

Example 1: The legs of a right triangle have lengths $a = 3$ cm and $b = 4$ cm. Compute the length of the hypotenuse.

c^{2} = a^{2} + b^{2} = 3^{2} + 4^{2} = 9 + 16 = 25

c = 25 = 5 cm

Example 2: $a = 9$ cm, $b = 12$ cm. Compute $c$ .

c^{2} = 9^{2} + 1 2^{2} = 81 + 144 = 225

c = 225 = 15 cm

Example 3: $a = 8$ cm, $b = 15$ cm. Compute $c$ .

c^{2} = 8^{2} + 1 5^{2} = 64 + 225 = 289

c = 289 = 17 cm

Example 1: The hypotenuse of a right triangle has length $c = 13$ cm and one leg is $a = 5$ cm. Compute the other leg $b$ .

b^{2} = c^{2} - a^{2} = 1 3^{2} - 5^{2} = 169 - 25 = 144

b = 144 = 12 cm

Example 2: $c = 25$ cm, $a = 7$ cm. Compute $b$ .

b^{2} = 2 5^{2} - 7^{2} = 625 - 49 = 576

b = 576 = 24 cm

Example 3: $c = 10$ cm, $b = 6$ cm. Compute $a$ .

a^{2} = c^{2} - b^{2} = 1 0^{2} - 6^{2} = 100 - 36 = 64

a = 64 = 8 cm

In practice, results often aren't whole numbers. In that case we round to a chosen number of decimal places.

Example 1: $a = 5$ cm, $b = 7$ cm. Compute $c$ (rounded to 2 decimal places).

c^{2} = 5^{2} + 7^{2} = 25 + 49 = 74

c = 74 \approx 8.60 cm

Example 2: $a = 4$ cm, $b = 9$ cm. Compute $c$ .

c^{2} = 16 + 81 = 97

c \approx 9.85 cm

Example 3: $c = 12$ cm, $a = 7$ cm. Compute $b$ .

b^{2} = 144 - 49 = 95

b \approx 9.75 cm

Example 1: Is a triangle with sides $5$ , $12$ , $13$ a right triangle?

The longest side is $13$ , so it should be the hypotenuse.

5^{2} + 1 2^{2} = 25 + 144 = 169 = 1 3^{2} ✓

Yes, it is a right triangle.

Example 2: Is a triangle with sides $4$ , $5$ , $7$ a right triangle?

The longest side is $7$ .

4^{2} + 5^{2} = 16 + 25 = 41

7^{2} = 49

41 \neq = 49

No, it is not a right triangle.