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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.


3D1


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.


3D2


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.


3D3


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.


3D4


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.


3D5


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.


3D6


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.

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