Pythagorean Theorem Calculation Examples

When you are working with the Pythagorean Theorem, there are two main types of problems you will face. Finding the length of the hypotenuse, and finding the length of a missing leg. These two possibilities are shown in the two examples below.

Once you have looked over these examples, you can attempt the sample questions.

Example 1 - Finding the Length of the Hypotenuse

Given the right triangle below, calculate the length of the hypotenuse. (Round your answer to one decimal place.)

Solution

Seeing as this IS a right-angled triangle, and we are given two sides to find the length of the third side, we will use the pythagorean theorem.

The first step is to substitute the length of the sides given into the formula:

(8.2)² + (4.3)² = c²

Now square the lengths of each side and simplify. (To square a number, multiply the number by itself.) Notice that we will NOT round off until we reach our FINAL answer.

67.24 + 18.49 = c²
85.73 = c²

The final step is to square root both sides of the equation to eliminate the exponent from the c variable. This will give us our final answer.

9.259 = c

Rounding to one decimal place gives us an answer of 9.3 cm for the hypotenuse.

Example 2 - Finding the Length of a Leg

Find the length of the missing side of the triangle shown below:

Solution

This example is a little more difficult as we are solving for one of the legs of a right triangle. When you are examining this solution, note how it differs from the solution to example 1.

The first step is to substitute the lengths of the sides we are given into the Pythagorean Theorem. However, make sure you put the length of the hypotenuse where you see the variable c. (In the Pythagorean Theorem, the c ALWAYS represents the hypotenuse.)

8² + b² = 10²

It does not matter which variable you use for the 8. Either a or b works fine for the length of a leg. Now square the sides and simplify. Notice that the simplification is a little more complex in this example.

64 + b² = 100
b² = 100 - 64
b² = 36

We have to isolate the b variable in order to solve for it. We have done so, so now just square root both sides of the equation.

For this question, the length of the missing leg is 6 cm.

Now that you have seen the calculation examples, you can attempt the sample questions.