# Rectangular & Square Pyramids: Definition & Examples

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will take a tour of the ancient pyramids of Egypt. We will look at the mathematical aspects of these pyramids, focusing on square and rectangular pyramids, in particular. We will also see some examples of these shapes in the modern world around us.

## Pyramids

Congratulations! You've just won a tour through the pyramids of Egypt (virtual, of course)! Even better news? Our tour guide is a mathematician, so we get a really neat perspective on these amazing ancient creations. Well, what are we waiting for - let's embark on this journey together!

Okay, we're off! Our tour guide begins by explaining to us that pyramids are magnificent structures that were built around the estimated timeframe of 2700 BC to 1500 BC and that over 100 of these pyramids still stand today. These pyramids were built by ancient Egyptians to serve as tombs for the pharaohs and their queens. Mathematically speaking, pyramids are three-dimensional objects that have the following properties.

• The bottom side of the pyramid is a polygon.
• The other sides of the pyramid are triangles, and they meet at the top of the pyramid.

We call the polygonal bottom side of the pyramid the base of the pyramid, and we call the other triangular sides the faces of the pyramid. Lastly, we call the point at which the triangular faces meet at the top of the pyramid the apex.

We take a pause to admire an especially large pyramid. Pretty amazing, isn't it? The tour guide goes on to tell us that this particular pyramid is one of the great pyramids of Giza, and that what's even more amazing about all of these pyramids is that they were all built by hand! Back when these were created, they didn't have the machinery we do nowadays. It is estimated that it took about 20,000 to 30,000 workers to build this specific pyramid we're admiring. Not only that, but it took them just shy of 23 years to build it! Wow!

We stop to ask the tour guide if, mathematically speaking, there is anything that sets a particular pyramid apart from the others. The tour guide gets excited and says he is so glad you asked, and asks you to take a look at the base of the pyramid. What is its shape? You answer that it looks like a rectangle or a square. The tour guide smiles and says yes! You are absolutely correct, and this is what actually sets any pyramid apart from the next - the shape of its base.

For instance, if a pyramid has a triangle for its base, we call it a triangular pyramid If a pyramid has a square for its base, we call it a square pyramid. If a pyramid has a rectangle as its base, we call it a rectangular pyramid. Are you seeing a pattern here? We name a pyramid based on the shape of its base.

You notice that most of the pyramids have a rectangular or square base, and you mention it to the tour guide. He says that this is the classic and most common type of pyramid that was built in ancient Egypt, so let's take a closer look at these types of pyramids, specifically.

## Square and Rectangular Pyramids

As we said, a pyramid that has a square as its base is called a square pyramid.

Now, technically, a square is a rectangle, because a rectangle is defined as a two-dimensional shape with four straight sides and four right angles. Well, a square fits this description. Therefore, we could also call a square pyramid a rectangular pyramid. Again, a rectangular pyramid is a pyramid with a rectangular base.

So, does that mean we can call all rectangular pyramids square pyramids as well? Well, let's think about it! A square is defined as a rectangle with four equal sides. Hmmm, well not all rectangles have equal sides, so if you're thinking that we can't necessarily call a rectangular pyramid a square pyramid, you are correct! Basically, it comes down to the fact that all squares are rectangles, so all square pyramids are rectangular pyramids. However, not all rectangles are squares, so not all rectangular pyramids are square pyramids.

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