Using the Pythagorean Theorem to Solve 3D Problems

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

The Pythagorean Theorem is a fascinating formula often used to solve two-dimensional problems. This lesson will explain how to extend this concept to three-dimensional problems in two ways.

Pythagorean Theorem

Suppose that an elementary school is building a new jungle gym for their playground. The builders creating the jungle gym have designed it so that it is the shape of a large rectangular prism.


After erecting the structure, the builders decide that they want to add two more metal bars, one as a diagonal of the front side of the structure and the other from the lower front corner to the upper back corner of the structure.


To do this, they need to find the length of the bars. Thankfully, we have a nice formula for doing so. The Pythagorean Theorem states that if a right triangle has side lengths a, b, and c, where c is the longest side (or the hypotenuse), then the following formula holds:

  • a2 + b2 = c2

Notice that the added bar that is on the front rectangle of the structure forms a right triangle with the two sides of that rectangle. The added bar is the hypotenuse and the other two side lengths are 3 meters and 4 meters. Therefore, if we let a = 3, b = 4, and c be the added bar length, we can use the Pythagorean theorem to find c.


We see that the metal bar that is the diagonal of the front rectangle will need to have length 5 meters. That was pretty easy! This is an example of using the Pythagorean Theorem for a two-dimensional problem.

What about the other bar? This one is part of the whole structure, so we are working in three dimensions now. We can still use the Pythagorean Theorem. We will just have to extend it to three dimensions, which isn't too hard to do!

Pythagorean Theorem in 3D Problems

One way of solving three-dimensional problems using the Pythagorean theorem is similar to two-dimensional problems, but we may need to use it more than once to find what we're looking for.

Consider the added bar that runs from the bottom front corner to the upper back corner of the structure. It's not a diagonal of an obvious rectangle, but it does form a right triangle with the diagonal of the bottom rectangle and the height of the structure.


We know the height of the structure is 3 meters, so we have one side of the right triangle. If we can find the length of the diagonal of the bottom rectangle, we would have two sides of the triangle and we could use the Pythagorean Theorem to find the length of the added metal bar. How can we find the length of that bottom rectangle's diagonal? If you're thinking the Pythagorean Theorem, then you're getting the idea!

We see that the bottom rectangle's diagonal forms a right triangle with the two sides of the bottom rectangle, and we know those side lengths are 4 meters and 2 meters. Therefore, we plug a = 4 and b = 2 into the Pythagorean Theorem and solve for c.


We get that the bottom rectangle's diagonal has a length of √(20). Awesome! Now, we can use the Pythagorean theorem again, but this time with a = 3 and b = √(20), to find the length of the added metal bar.


Ah-ha! We just needed to use the Pythagorean Theorem twice to get that the added metal bar running from the bottom front corner to the upper back corner of the jungle gym will have length √(29), or approximately 5.4 meters.

This all makes sense, but as it turns out, this process can be compacted down into just one step using the three-dimensional version of the Pythagorean Theorem.

To unlock this lesson you must be a Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use

Become a member and start learning now.
Become a Member  Back
What teachers are saying about
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account