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?

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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.

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