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Introduction

You may know that the volume of a prism is calculated by determining the area of the base and then multiplying that area by the number of layers required to build the height of the prism.

For the prism shown at right, the base has an area of 12 square units. Since the height of the prism is 5 units, there are 5 layers required to build the full prism, so the volume of the prism is 12 × 5 = 60 cubic units.

The area of a triangle is the area of a rectangle, the volume of a triangular prism is the volume of a rectangular prism with the same dimensions.

comparing rectangular and triangular prisms

Pyramid of Cestius in Rome, ItalyIn this lesson, you will extend what you may know about volume to pyramids. A pyramid is a three-dimensional figure that, unlike a prism, only has one polygonal base. The vertices of the base are joined by segments creating lateral faces shaped like triangles that meet at one point above the base of the pyramid. That point is called the vertex of the pyramid. You may also see that point called the apex of a pyramid.

For example, the Pyramid of Cestius, in Rome, Italy, is an example of a pyramid with a rectangular base. You can see in the picture that the lateral faces are triangles, and that the edges of the lateral faces all meet at one point at the top, or vertex, of the pyramid.

Source: 6276 - Roma - Piramide di Caio Cestio - Foto Giovanni Dall'Orto - 31-Mar-2008, Giovanni Dall'Orto, Wikimedia Commons

Comparing Pyramids and Prisms

Volume of Prisms

To begin this section, let's take a look back at how to determine the volume of prisms.

Click on the image below to access an interactive to investigate the formula for the volume of prisms. Follow the onscreen prompts until you reach the summary screen in the interactive.

Need additional directions?

image of the Find the Volume of a Rectangular Prism interactive

 

Volume of Pyramids

Now that you’ve written a general formula for the volume of any prism, let’s extend that to write a formula for the volume of a pyramid.

Recall that a prism and a pyramid both have polygonal bases. Let’s consider a prism and a pyramid that have congruent bases and the same height as shown in the image below.

diagram showing a pyramid and prism with congruent bases and the same height

Use the applet below to pour the volume of one pyramid into a prism with a congruent base and the same height. In the applet, select the “Cone -> Tank” option, and click “New Problem” until you have a pyramid and a prism. Use the slider to estimate the height of the liquid inside the prism once it is poured from the pyramid into the prism. Then, click “Pour” to pour the liquid and check your estimate. Repeat this for three or four different pyramids until you see a pattern.

image of the How High? applet


Need additional directions?

 

Pause and Reflect

 

Let’s suppose a prism and pyramid have congruent bases and congruent heights.

1. If you were to fill the pyramid with water and empty it into the prism, how many pyramids would it take to completely fill the prism?

2. How is the volume formula for a pyramid related to the volume formula for a prism?

 

Solving Problems Involving the Volume of Rectangular Pyramids

V equals one-third B hThe volume formula for any pyramid relates the volume, V, to the area of the base, B, and the height of the prism, h.

The way that you might choose to approach a volume problem involving a pyramid depends on the shape of the base. In this section, you will focus on rectangular pyramids, or pyramids with rectangular bases.

Cheops Pyramid, Giza, Egypt

 

The Cheops Pyramid in Egypt is a square pyramid, or a pyramid whose base is a square. Ancient Egyptians constructed the Cheops Pyramid.


 

 

Kukulcan Pyramid, Chichen Itza, Mexico

 

 

The Kukulcan Pyramid at Chichen Itza, Mexico, is also an example of a square pyramid. Ancient Mayans constructed the Kukulcan Pyramid.


 


Fluorite pyramid and blue laser

A pyramid of fluorite, a mineral found throughout the world, can be used to refract the light from a blue laser and create special visual effects!

Rectangular pyramids appear frequently in art and architecture. Because they are so common, it is important to be able to calculate the volume of a rectangular pyramid.

A problem-solving process may help you when you encounter problems related to a rectangular pyramid.

 

image of the four step problem solving process

Image Sources:
Flickr - Gaspa - Giza, la piramide grande, Francesco Gasparetti, Wikimedia Commons
Kukulcan, Chichén Itzá, Kyle Simourd, Wikimedia Commons
Fluorite pyramid and blue laser, Mike Lewinski, Flickr Commons

 

Consider the problem shown below.

square pyramid with dimensions


See how Sabrina used a problem-solving model to solve this problem in the interactive below.

Pause and Reflect

If you were solving the same problem as Sabrina, what solution strategy would you have selected?

Sabrina used rounding to check her answer for reasonableness. What is another way to know that your answer is reasonable?

 

 

Solving Problems Involving the Volume of Triangular Pyramids

Pyramids are named by the shape of their bases. Rectangular pyramids have rectangles for bases. Triangular pyramids have triangles for bases.In the last section, you calculated the volume of a rectangular pyramid using the formula, V = Bh, where B represents the area of the base of the pyramid, and h represents the height of the pyramid.

However, as you have seen, not all pyramids are square or rectangular. In this section, you will focus on calculating the volume of triangular pyramids, which are pyramids whose bases are triangles.

 

Parks and Recreation

The Parks and Recreation department of a local city wants to build a veterans monument in the shape of a triangular pyramid. The monument cannot exceed a volume of 500 cubic feet. Three designs have been submitted for consideration.

Three triangular pyramid designs

Which of the submitted designs meets the maximum volume requirement?

To solve this problem, use the four-step problem-solving model.

Pause and Reflect

How does the four-step problem-solving process help you to solve volume problems?

How is calculating the volume of a triangular pyramid different from calculating the volume of a rectangular pyramid?

 

Summary

In this lesson, you solved problems involving the volume of rectangular pyramids and triangular pyramids. As you noticed, there are also many other types of pyramids. However, you will learn more about determining the volumes of those pyramids in later courses.

comparing rectangular pyramids and triangular pyramids

Pyramids are important figures because they are very common in art and architecture. For example, the Louvre museum in Paris contains a glass pyramid above the visitors center.

Glass pyramid at the Louvre in Paris

Source: Louvre (12621417763), Miguel Mendez, Wikimedia Commons