Debug
demonstration of the pythagorean theorem

Pythagorean Theorem

How to Use The Pythagorean Theorem

The Formula

The picture below shows the formula for the Pythagorean theorem. For the purposes of the formula, side $$ \overline{c}$$ is always the hypotenuse. Remember that this formula only applies to right triangles.

The Pythagorean Theorem

Examples of the Pythagorean Theorem

When you use the Pythagorean theorem, just remember that the hypotenuse is always 'C' in the formula above. Look at the following examples to see pictures of the formula.

Examples of the Pythagorean Theorem

Video Tutorial

on How to Use the Pythagorean Theorem

Step By Step Examples

of Using the Pythagorean Theorem

Example 1 (solving for the hypotenuse)

Use the Pythagorean theorem to determine the length of X.

Example 1 Step 1

Identify the legs and the hypotenuse of the right triangle.

The legs have length 6 and 8. $$X $$ is the hypotenuse because it is opposite the right angle.

The hypotenuse is red in the diagram below: 3, 4, 5 right triangle
Step 2

Substitute values into the formula (remember 'C' is the hypotenuse).

$ A^2+ B^2= \red C^2 \\ 6^2+ 8^2= \red X^2 $

Step 3
Solve for the unknown.

$A^2+ B^2= \red X^2 \\ 100= \red X^2 \\ \sqrt {100} = \red X \\ 10= \red X $

Example 2 (solving for a Leg)

Use the Pythagorean theorem to determine the length of X.

Example 2 Step 1

Identify the legs and the hypotenuse of the right triangle.

The legs have length 24 and $$X$$ are the legs. The hypotenuse is 26.

The hypotenuse is red in the diagram below: 10 24 25  right triangle
Step 2

Substitute values into the formula (remember 'C' is the hypotenuse).

$ \red A^2+ B^2= C^2 \\ \red x^2 + 24^2= {26}^2 $

Step 3
Solve for the unknown.

$ \red x^2 + 24^2= 26^2 \\ \red x^2 + 576= 676 \\ \red x^2 = 676 - 576 \\ \red x^2 = 100 \\ \red x = \sqrt { 100} \\ \red x = 10 $

Practice Problems

Problem 1

Find the length of X.

Pythagorean Theorem Problem

Remember our steps for how to use this theorem. This problems is like example 1 because we are solving for the hypotenuse .

Step 1

Identify the legs and the hypotenuse of the right triangle.

The legs have length 14 and 48. The hypotenuse is X.

Step 2

Substitute values into the formula (remember 'C' is the hypotenuse).

$ A^2 + B^2 = C^2 \\ 14^2 + 48^2 = x^2 $

Step 3

Solve for the unknown.

$ 14^2 + 48^2 = x^2 \\ 196 + 2304 = x^2 \\ \sqrt{2500} = x \\ \boxed{ 50 = x} $

The hypotenuse is red in the diagram below: right triangle 7, 24, 25
Problem 2

Use the Pythagorean theorem to calculate the value of X. Round your answer to the nearest tenth.

Pythagorean Theorem Problem

Remember our steps for how to use this theorem. This problems is like example 2 because we are solving for one of the legs.

Step 1

Identify the legs and the hypotenuse of the right triangle.

The legs have length 9 and X. The hypotenuse is 10.

Step 2

Substitute values into the formula (remember 'C' is the hypotenuse).

$ A^2 + B^2 = C^2 \\ 9^2 + x^2 = 10^2 $

Step 3

Solve for the unknown.

$ 9^2 + x^2 = 10^2 \\ 81 + x^2 = 100 \\ x^2 = 100 - 81 \\ x^2 = 19 \\ x = \sqrt{19} \approx 4.4 $

Problem 3

Use the Pythagorean theorem to calculate the value of X. Round your answer to the nearest hundredth.

Pythagorean Theorem Problem

Remember our steps for how to use this theorem. This problems is like example 2 because we are solving for one of the legs.

Step 1

Identify the legs and the hypotenuse of the right triangle.

The legs have length '10' and 'X'. The hypotenuse is 20.

Step 2

Substitute values into the formula (remember 'C' is the hypotenuse).

$ A^2 + B^2 = C^2 \\ 10^2 + \red x^2 = 20^2 $

Step 3

Solve for the unknown.

$ 10^2 + \red x^2 = 20^2 \\ 100 + \red x^2 = 400 \\ \red x^2 = 400 -100 \\ \red x^2 = 300 \\ \red x = \sqrt{300} \approx 17.32 $


Back to Triangles Next to Pythag Theorem Shell Problem