# Lesson 20

The Volume of a Sphere

### Lesson Narrative

The purpose of this lesson is for students to recognize that the volume of a sphere with radius \(r\) is \(\frac43 \pi r^3\) and begin to use the formula. Students inspect an image of a sphere that snugly fits inside a cylinder (they each have the same radius, and the height of the cylinder is equal to the diameter of the sphere), and use their intuition to guess about how the volume of the sphere relates to the volume of the cylinder, building on the work in the previous lesson. Then, they watch a video that shows a sphere inside a cylinder, and the contents of a cone (with the same base and height as the cylinder) are poured into the remaining space. This demonstration shows that for these figures, the cylinder contains the volumes of the sphere and cone together. From this observation, the volume of a specific sphere is computed. Then, the formula \(\frac43 \pi r^3\) for the volume of a sphere is derived. (At this point, this is taken to be true for any sphere even though we only saw a demonstration involving a particular sphere, cone, and cylinder. A general proof of the formula for the volume of a sphere would require mathematics beyond grade level.)

### Learning Goals

Teacher Facing

- Calculate the volume of a sphere, cylinder, and cone which have a radius of $r$ and height of $2r$, and explain (orally) the relationship between their volumes.
- Create an equation to represent the volume of a sphere as a function of its radius, and explain (orally and in writing) the reasoning.

### Student Facing

Let’s explore spheres and their volumes.

### Required Preparation

For the A Sphere in a Cylinder activity, students will need to view a video.

### Learning Targets

### Student Facing

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