How to find missing side of right triangle

Question

The Pythagorean theorem states that a2 + b2 = c2 in a right triangle where c is the longest side. You can use this equation to figure out the length of one side if you have the lengths of the other two. The figure shows two right triangles that are each missing one side's measure.

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Defining the Hypotenuse
How the Pythagorean Theorem Applies
Summary for Finding the Hypotenuse
More Math Homework Help
How do you find the third missing side of a right triangle?

In the left triangle, the measure of the hypotenuse is missing. Use the Pythagorean theorem to solve for the missing length.

Replace the variables in the theorem with the values of the known sides.
482 + 142 = c2
Square the measures and add them together.

The length of the missing side, c, which is the hypotenuse, is 50.

The triangle on the right is missing the bottom length, but you do have the length of the hypotenuse. It doesn't matter whether you call the missing length a or b.

Replace the variables in the theorem with the values of the known sides.

332 + b2 = 1832
Square the measures, and subtract 1,089 from each side.
Find the square root of each side.

The length of the missing side is 180 units. That's not much shorter than the hypotenuse, but it still shows that the hypotenuse has the longest measure.

About This Article

This article is from the book:

Trigonometry For Dummies ,

About the book author:

Mary Jane Sterling is the author of Algebra I For Dummies and many other For Dummies titles. She has been teaching mathematics at Bradley University in Peoria, Illinois, for more than 30 years and has loved working with future business executives, physical therapists, teachers, and many others.

This article can be found in the category:

Trigonometry ,

The Pythagorean Theorem is an important theorem, as it comes up often in high school math. It makes the question of how to find the hypotenuse of a right triangle is easy to answer.

Plus, we can use it to find the base of a special right triangle and apply the converse of the Pythagorean Theorem. Let’s review this basic but foundational concept in trigonometry and how it relates to finding the hypotenuse of a right triangle.

Defining the Hypotenuse

First, let's demystify this term. "Hypotenuse" is simply a term that means "the longest side of a right triangle." The hypotenuse is the opposite side of the right angle in the triangle. It’s also the longest side of the triangle.

If a problem asks you to calculate the length of hypotenuse c in a triangle with side a, side b, and hypotenuse c, then you are working with a right-angled triangle.

How the Pythagorean Theorem Applies

The Pythagorean Theorem is a well-known theorem developed by a Greek mathematician named Pythagoras around 500 BC. It related the length of the hypotenuse of a right triangle to the lengths of the legs (the legs of the triangle are the two shorter sides). The Pythagorean Theorem states that for any right triangle, the sum of the squares of the lengths of the legs is always equal to the square of the length of the hypotenuse.

The Pythagorean Theorem states that:

Because of the Pythagorean Theorem, it is easy to find the hypotenuse of a right triangle if we are given the sides of a right triangle. In fact, if we know the lengths for any two sides (leg A, leg B, and hypotenuse C) we can easily figure out the missing side by applying the formula.

This can help you find any missing side. But for now, let's see an example where we know the length of the sides and want to find the hypotenuse:

Here we know the length of the side (a = 3 and b = 4). If we substitute it into the formula, we get:

And by taking the square root on both sides, we figure out that the hypotenuse c = 5!

Summary for Finding the Hypotenuse

The Pythagorean Theorem helps us calculate the hypotenuse of a right triangle if we know the sides of the triangle.

There are more advanced trigonometric functions that allow us to calculate the third side of a triangle, even non-right triangles, given a particular degree angle and side length.

But either way, practice applying the Pythagorean Theorem until you feel confident with right triangles. Then, you can learn how to find the third side of any triangle.

More Math Homework Help

What Is the Converse of the Pythagorean Theorem?
How To Find the Base of a Triangle in 4 Different Ways
How You Use the Triangle Proportionality Theorem Every Day
Learn To Find the Area of a Non-Right Triangle

How do you find the third missing side of a right triangle?

The hypotenuse is opposite the right angle and can be solved by using the Pythagorean theorem. In a right triangle with cathetus a and b and with hypotenuse c , Pythagoras' theorem states that: a² + b² = c² . To solve for c , take the square root of both sides to get c = √(b²+a²) .

find missing side triangle