The Volume Formula of a Sphere

New York State Common Core Math Geometry, Module 3, Lesson 12

Worksheets for Geometry

Student Outcomes

• Students give an informal argument using Cavalieri’s principle for the volume formula of a sphere and use the volume formula to derive a formula for the surface area of a sphere.

The Volume Formula of a Sphere

Classwork

Opening Exercise

Picture a marble and a beach ball. Which one would you describe as a sphere? What differences between the two could possibly impact how we describe what a sphere is?

• Show Video

Example

Use your knowledge about the volumes of cones and cylinders to find a volume for a solid hemisphere of radius R.

Exercises

1. Find the volume of a sphere with a diameter of 12 cm to one decimal place.
2. An ice cream cone is 11 cm deep and 5 cm across the opening of the cone. Two hemisphere-shaped scoops of ice cream, which also have diameters of 5 cm, are placed on top of the cone. If the ice cream were to melt into the cone, would it overflow?
3. Bouncy rubber balls are composed of a hollow rubber shell 0.4" thick and an outside diameter of 1.2". The price of the rubber needed to produce this toy is $0.035/in³. a. What is the cost of producing 1 case, which holds 50 such balls? Round to the nearest cent. b. If each ball is sold for $0.10, how much profit is earned on each ball sold?

Lesson Summary

SPHERE: Given a point 𝐶 in the three-dimensional space and a number 𝑟 > 0, the sphere with center 𝐶 and radius 𝑟 is the set of all points in space that are distance 𝑟 from the point 𝐶.

SOLID SPHERE OR BALL: Given a point 𝐶 in the three-dimensional space and a number 𝑟 > 0, the solid sphere (or ball) with center 𝐶 and radius 𝑟 is the set of all points in space whose distance from the point 𝐶 is less than or equal to 𝑟.

Check out our most popular games!

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.