Introduction

The volume of a 3-dimensional solid is another important attribute. Volume is the amount of space that a 3-dimensional object takes up.

In this lesson, you will examine different ways to find the volume of 3-dimensional objects, including prisms, cylinders, pyramids, cones, and spheres.

Prisms are 3-dimensional figures with congruent bases that are polygons and that have lateral faces in the shapes of parallelograms.

Cylinders are 3-dimensional figures with congruent bases that are circles and have a curved lateral surface that unfolds to be the shape of a rectangle.

Pyramids are 3-dimensional figures with only one base that is a polygon. The lateral faces of a pyramid are triangles that all meet at the vertex of the pyramid.

Cones are 3-dimensional figures that, like pyramids, have only one base, but that base is a circle. The lateral surface of a cone is curved and comes to one point at the top of the cone, called the vertex of the cone.

Spheres are 3-dimensional figures with one completely round surface. Every point on the surface of a sphere is the same distance from the center of the sphere, so a sphere looks like a round ball.

The volumes of these 3-dimensional figures are related in special ways. You will investigate those relationships in this lesson, as well as examine different ways to determine or express the volume of 3-dimensional figures.

Expressing Volumes of Prisms and Pyramids

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 the 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?

  1. How many unit cubes did it take to form the bottom layer of the cube?
  2. How many of these layers would it take to fill the entire cube?
  3. In total, how many of the unit cubes were needed to fill the cube or polyhedron?
  4. How did you determine the area of the base layer?
  5. How did you determine the total number of cubes it took to fill the large cube?

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

The images below illustrate pouring the volume of one pyramid into a prism with a congruent base and the same height.

1. If the height of the prism is 10 units, about how high will the liquid from the pyramid fill the prism?

2. In general, for any height of prism and pyramid, about how high will the liquid from the pyramid fill the prism?

Practice

Expressing Volumes of Cylinders, Cones, and Spheres

In the previous section, you looked for relationships between the volume formulas for prisms and pyramids. In this section, you will investigate relationships between cylinders, cones, and spheres.

Volume of Cylinders

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

Need additional directions?

  1. What was the area of the circular base of the cylinder?
  2. What was the height of the height of the cylinder?
  3. How did you determine the area of the base?
  4. What was the volume of the cylinder?
  5. How is the volume of a cylinder related to the area of the base of the cylinder and the height of the cylinder?

Volumes of Cones and Spheres

Cones and spheres have a special relationship to cylinders. As with pyramids and prisms, let’s begin by thinking about a cylinder, cone, and sphere with congruent dimensions.

Cylinder with radius equal to r and height equal to 2r, sphere with radius equal to r, and cone with radius equal to r and height equal to 2r.

The images below show the same cylinder and cone from the diagram. Use the series of images to see how many times you need to fill the cone and pour it into the cylinder before you have a completely full cylinder.

  1. How many times did you need to fill the cone and pour it into the cylinder in order to completely fill the cylinder?
  2. If the formula for the volume of a cylinder is V = 13\frac{1}{3}πr2h, what is the formula for the volume of a cone?
  3. If the “height” of the sphere is equal to the diameter (2 times the radius, or 2r), how does that change your volume formula for a sphere?

The images below show the same cone and sphere from the diagram. Use the series of images to see how many times you need to fill the cone and pour it into the sphere before you have a completely full sphere.

  1. How many times did you need to fill the cone and pour it into the sphere in order to completely fill the sphere?
  2. If the formula for the volume of a cone V = 13\frac{1}{3}πr2h, what is the formula for the volume of a sphere?
  3. If the “height” of the sphere is equal to the diameter (2 times the radius, or 2r), how does that change your volume formula for a sphere?

Using Nets to Determine Volume

You may also see nets used as a representation of 3-dimensional figures in problems that ask you to determine the volume of the resulting figure.

The Candy Shop

Mr. McCallum works at a candy shop and is designing a new box that will hold small pieces of chocolates. The net for the box, which will be stamped out of paperboard, has the following design:

Net of a rectangular prism with dimensions labeled”

In this design, the faces that are shaded orange will be the bases of the box that, when it is folded together, will be in the shape of a rectangular prism.

Use the formula you identified earlier, V = Bh, to write the expression. What are the dimensions of the base of the rectangular prism? Click the box beneath each dimension to select the number.

  1. Write an expression that you can use to determine the area of the base of the prism.
  2. Write an expression that you can use to determine the area of the volume of the prism.
  3. Use the expression to calculate the volume of the candy box.

Practice

Write an expression that you can use to determine the volume of each of the figures that is represented by the nets below.

Triangular Prism

Net of a triangular prism with dimensions labeled

Cylinder

Net of a cylinder with dimensions labeled

Summary