Sphere Volume

Key Concepts

• Find the volume of a sphere.
• Find the volume of a composite solid.

Introduction

Volumes of Spheres

Imagine that the interior of a sphere with radius r is approximated by n pyramids as shown, each with a base area of B and a height of r, as shown. The volume of each pyramid is 1/3 Br, and the sum is nB. 

V » n(1/3)Br (Each pyramid has a volume of 1/3Br) 

   = 1/3 (nB)r (Regroup factors) 

   » 1/3(4πr²)r (Substitute 4πr² for nB) 

   = 4/3πr² (Simplify) 

Volume of a sphere = 4(𝛑)( radius )³/𝟑

Find the volume of a sphere 

Example 1: 

Find the volume of the given sphere. 

Solution: 

The volume V of a sphere is V = 4/3πr³ , where r is the radius. 

The radius is 1 cm. 

V = 4/3πr³

≈ 4.2 cm³  

The volume of the sphere is 4.2 cm³. 

Example 2: 

The Reunion Tower in Dallas, Texas, is topped by a spherical dome that has a surface area of approximately 13,924π square feet. What is the volume of the dome? Round to the nearest tenth. 

Solution: 

Find r. 

S = 13,924π

4πr² = 13,924π

4r² = 13,924

r² = 3481

r = 59

Find the volume. 

V = 4/3π(59)³

≈ 860,289.5ft³  

The volume of the Reunion Tower is 860,289.5 ft³. 

Find the volume of a composite solid 

Example 3: 

Find the volume of the composite solid. 

Solution: 

To find the volume of the figure, first, we need to calculate the volume of the cylinder and then the volume of the hemisphere and add both volumes. 

Volume of the cylinder is, 

V(cylinder) = πr²h

= π(4)2(5)

= 80π

The volume of the hemisphere is, 

V(hemisphere) = 2/3πr³

= 2/3π(4)³

= 128π/3

The volume of the composite solid is, 

V(figure) = cylinder + hemisphere

= 80π+128π/3

≈ 385.4in³

The volume of the composite solid is 385.4 in³. 

Example 4: 

Find the volume of the composite solid. 

Solution: 

The volume of the figure is the volume of the prism minus the volume of the hemisphere. 

Volume of the prism is, 

Square prism volume: V = a²h 

    = 10² × 13 

    = 1300 

Volume of the hemisphere is, 

V(hemisphere) = 2/3πr³

= 2/3π(5)³

= 250π/3

Volume of the composite solid is, 

V(figure) = prism – hemisphere

= 1300 + 250π/3

≈ 1038.2

The volume of the composite solid is 1038.2 cm³. 

Exercise

• Find the volume of the sphere using the given radius r.

• Find the volume of the sphere using the given diameter d.

• Find the volume of the sphere using the given radius r.

• Find the volume of the hemisphere using the given radius r.

• The volume of a sphere is 36π cubic feet. What is the diameter of the sphere?
• The volume of a sphere is 300 ft³. Find the radius.
• The circumference of a tennis ball is 8 inches. Find the volume of a tennis ball.
• Tennis balls are stored in a cylindrical container with a height of 8.625 inches and a radius of 1.43 inches. The circumference of a tennis ball is 8 inches. There are 3 tennis balls in the container. Find the amount of space within the cylinder not taken up by the tennis balls.

• Find the volume of the composite solid.

• Find the volume of the composite solid.

Concept Map

What have we learned

• Calculate the volume of a sphere using volume formula.
• Calculate the volume of a composite solid by adding the volumes of two or more solids.