Volume, Faces & Vertices of an Octagonal Pyramid

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  • 0:04 Octagonal Pyramid
  • 1:27 Volume of an Octagonal Pyramid
  • 3:10 Another Example
  • 3:53 Lesson Summary
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Lesson Transcript
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.

Octagonal pyramids show up often in architecture and engineering. This lesson defines these pyramids and their parts. We'll then look at the volume formula for octagonal pyramids and show, through examples, how to use it to find the volume of these types of pyramids.

Octagonal Pyramid

Suppose you just decided to have a solar panel sunroof unit installed on your roof to save energy and money on heating bills. The unit is made of all glass, and the bottom side is in the shape of an octagon, where an octagon is an eight-sided polygon. The sides of the unit are triangles that connect to each of the sides of the bottom and meet at a point directly above the bottom.

In mathematics, we call the shape of this unit an octagonal pyramid. An octagonal pyramid is a pyramid that has a bottom that's the shape of an octagon and has triangles as sides. These types of pyramids have nine sides all together, called faces. The bottom face (the octagon) is called the base, and the other eight faces are the sides of the pyramid that all meet at a single point directly above the base, forming the pyramid.

The corners at which the edges of each of the faces meet are called the vertices of the pyramid. An octagonal pyramid has nine vertices; eight are located where the triangular faces meet the base and the ninth is the point at which all of the triangular faces meet at the top of the pyramid. We often call the vertex on top the apex of the pyramid.

Lastly, the height of an octagonal pyramid is the length of the line segment that's perpendicular to the base of the pyramid and which runs through the apex of the pyramid.


Phew! That's a lot of definitions! Who knew that there was so much to be said about this little solar panel?

Volume of an Octagonal Pyramid

Once your solar panel is delivered, you want to know how much space is inside of the panel. After all, more space means more energy saved. Thankfully, we have a nice formula for finding the volume of an octagonal pyramid. To use the formula, we simply need to know the length of one of the sides of the base of the pyramid and the height of the pyramid. Once we have these facts, we can use the following formula to find the volume of the pyramid.

  • Volume = (B × h) / 3, where B is the area of the base
  • B = 2 × s2 (1 + √2)

Therefore, in order to find the volume of an octagonal pyramid, we first find the area of the base, then we plug that value and the height of the pyramid into the volume formula. Okay, that's not too bad. The area of the base formula is a bit involved, but it all comes down to plugging in values and simplifying. We can do this!

You take some measurements of your solar panel, and you find that the height of your solar panel is 8 feet, and the length of one of the sides of the base is 3 feet. Let's figure out this volume!

First, we find the area of the base by plugging in the side length of s = 3 into the base area formula, and then we simplify.


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