Lesson 20The Volume of a Sphere

Learning Goal

Let’s explore spheres and their volumes.

Learning Targets

  • I can find the volume of a sphere when I know the radius.

Warm Up: Sketch a Sphere

Problem 1

Here is a method for quickly sketching a sphere:

  • Draw a circle.

  • Draw an oval in the middle whose edges touch the sphere.

Two figures that demonstarte how to sketch a sphere. The first image is a circle. An arrow points from the first image to the second image. The second image is the image of the first circle, with a horizontal oval drawn in the middle whose edges touch the sphere. The "front" half of the oval is a solid line, the "back" half of the oval is a dashed line.

  • Practice sketching some spheres. Sketch a few different sizes.

  • For each sketch, draw a radius and label it .

Activity 1: A Sphere in a Cylinder

Problem 1

An image with three shapes: a cone, a sphere and a cylinder.

Here are a cone, a sphere, and a cylinder that all have the same radii and heights. The radius of the cylinder is 5 units. When necessary, express all answers in terms of .

  1. What is the height of the cylinder?

  2. What is the volume of the cylinder?

  3. What is the volume of the cone?

  4. What is the volume of the sphere? Explain your reasoning.

Activity 2: Spheres in Cylinders

Problem 1

An image with three shapes: a cone, a sphere and a cylinder.

Here are a cone, a sphere, and a cylinder that all have the same radii and heights. Let the radius of the cylinder be units. When necessary, express answers in terms of .

  1. What is the height of the cylinder in terms of ?

  2. What is the volume of the cylinder in terms of ?

  3. What is the volume of the cone in terms of ?

  4. What is the volume of the sphere in terms of ?

  5. A volume of the cone is the volume of a cylinder. The volume of the sphere is what fraction of the volume of the cylinder?

Lesson Summary

Think about a sphere with radius units that fits snugly inside a cylinder. The cylinder must then also have a radius of units and a height of units. Using what we have learned about volume, the cylinder has a volume of , which is equal to cubic units.

We know from an earlier lesson that the volume of a cone with the same base and height as a cylinder has of the volume. In this example, such a cone has a volume of or just cubic units.

A cone, a sphere and a cylinder. The height of the cone is labeled "2 r" and the radius of the cone is labeled "r." The radius of the sphere is labeled "r." The height of the cylinder is labeled "2 r" and the radius of the cylinder is labeled "r."

If we filled the cone and sphere with water, and then poured that water into the cylinder, the cylinder would be completely filled. That means the volume of the sphere and the volume of the cone add up to the volume of the cylinder. In other words, if is the volume of the sphere, then

This leads to the formula for the volume of the sphere,