Lesson 20: The Volume of a Sphere
Let’s explore spheres and their volumes.
Illustrative Math Unit 8.5, Lesson 20 (printable worksheets)
Lesson 20 Summary
The following diagram explores spheres and their volumes.
Lesson 20.1 Notice and Wonder: Two Shapes
Here is a method for quickly sketching a sphere:
- Draw a circle.
- Draw an oval in the middle whose edges touch the sphere.
- Practice sketching some spheres. Sketch a few different sizes.
- For each sketch, draw a radius and label it .
Lesson 20.2 A Sphere in 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 π.
- What is the height of the cylinder?
- What is the volume of the cylinder?
- What is the volume of the cone?
- What is the volume of the sphere? Explain your reasoning.
Lesson 20.3 Estimating Hemispheres
- Here are a cone, a sphere, and a cylinder that all have the same radii and heights. Let the radius of the cylinder be r units. When necessary, express answers in terms of π.
- What is the height of the cylinder in terms of r?
- What is the volume of the cylinder in terms of r?
- What is the volume of the cone in terms of r?
- What is the volume of the sphere in terms of r?
- A volume of the cone is 1/3 the volume of a cylinder. The volume of the sphere is what fraction of the volume of the cylinder?
Lesson 20 Practice Problems
- Match the description of each sphere to its correct volume.
- a. A cube’s volume is 512 cubic units. What is the length of its edge?
b. If a sphere fits snugly inside this cube, what is its volume?
c. What fraction of the cube is taken up by the sphere? What percentage is this? Explain or show your reasoning.
- Sphere A has radius 2 cm. Sphere B has radius 4 cm.
a. Calculate the volume of each sphere.
b. The radius of Sphere B is double that of Sphere A. How many times greater is the volume of B?
- Three cones have a volume of 192π cm3. Cone A has a radius of 2 cm. Cone B has a radius of 3 cm. Cone C has a radius of 4 cm. Find the height of each cone.
- The graph represents the average price of regular gasoline in the United States in dollars as a function of the number of months after January 2014.
a. How many months after January 2014 was the price of gas the greatest?
b. Did the average price of gas ever get below $2?
c. Describe what happened to the average price of gas in 2014.
- While conducting an inventory in their bicycle shop, the owner noticed the number of bicycles is 2 fewer than 10 times the number of tricycles. They also know there are 410 wheels on all the bicycles and tricycles in the store. Write and solve a system of equations to find the number of bicycles in the store.
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