Volume Of Spheres – Concept and Explanation

Key Concepts

• Relating volumes of cones and spheres.
• Finding the volume of sphere given surface area of the sphere.

Introduction: 

Sphere: 

A sphere is the set of points in space that are a fixed distance from a point called the center of the sphere. The intersection of a sphere and a plane that contains the center of the sphere is a great circle. A great circle divides a sphere into two congruent halves that are called hemispheres. 

Volume of a Sphere: 

The volume V of a sphere is  

V=4/3πr ³ 

Where r is the radius of the sphere. 

8.4.1 Relating Volumes of Cones and Spheres 

The volume of a sphere is the same as twice the volume of a cone with the same circular base and height. 

The height of a sphere is twice its radius. 

Example 1: 

What is the amount of air,in cubic centimeters, needed to fill the stability ball? Use 3.14 for π and round to the nearest whole number. 

Solution: 

Use the formula V= 4/3πr ³ 

V=4/3π(27.5)³  

V=4/3π(20796.87) 

V=(83187.48/3)× 3.14 

V= 87069.56 

Example 2: 

A British thermal unit (BTU) is a unit of energy. It is approximately the amount of energy needed to increase the temperature of one pound of water by one degree Fahrenheit. As you will see in the following example, the energy content of a fuel may be measured in BTUs per unit of volume.  

A spherical gas tank has the dimensions shown. When filled with natural gas, it provides 275,321 BTU. How many BTUs does one cubic foot of natural gas yield? Round to the nearest BTU. 

Solution: 

Step 1: Find the volume of the sphere. Use 3.14 for π. 

V=4/3πr ³ 

V=4/3π(4)³  

V=4/3π(64) 

V=(256/3)× 3.14 

V= 267.95 

The volume of the spherical gas tank is approximately 267.95 cubic feet. 

Step 2: Find the number of BTUs contained in one cubic foot of natural gas. 

=

275,321267.95

= 1027.51 

8.4.2 Finding the Volume of Sphere Given Surface Area of the Sphere 

Example 3: 

What is the volume of the bowling ball, rounded to the nearest whole number? Use 3.14 for π. 

Solution: 

Step 1:  

A sphere represents the bowling ball. Find the radius of the bowling ball. 

S.A. = 4πr ² 

2,122.64 = 4πr ² 

2,122.64/4 × 3.14 = r ² 

169 = r ² 

13 = r 

The radius is about 13 centimeters. 

Step 2: 

Find the volume of the bowling ball. 

V= 4/3πr ³ 

V=4/3π(13)³  

V=4/3π(2197) 

V= (8788/3) × 3.14 

V= 9,198.10 

The volume of the bowling ball is approximately 9,198.10 cubic centimetres. 

8.4.3 Finding the Volume of Composite Figure 

A composite figure is the combination of two or more figures into one object. 

 Example 3: 

Amy purchased a new glass in the market and wants to find the volume of it. How can she calculate the volume? Use 3.14 for π. 

Solution: 

Step 1: Find the volume of the hemisphere. 

V=

1/2×4/3 π(4)³

V=1/2×4/3 (64)π 

V=128/3π 

The volume of the hemisphere is

128/3π cubic centimetres. 

Step 2: Find the volume of the cylinder. 

V = πr ² × h 

= π(4)² × 3 

= π(16) × 3 

=48π  

Step 3: Add the volumes. 

128/3π + 48π =

272/3 π 

= 90.67  

Exercise:

1. ____________________ is a curved surface having a center equally distant from all points on the surface.
2. Find the volume of sphere with a radius of 1.3 yds.
3. Clarissa has a decorative bulb in the shape of a sphere. If it has a radius of 3 inches, what is its volume? Use 3.14 for π.
4. The volume of a sphere is 1436.8 ft3; find the radius of the sphere.
5. A sphere has a surface area of about 803.84 square centimeters. What is the volume of the sphere? Use 3.14 for π and round to the nearest whole number.
6. A sphere has a surface area of about 2,826 square millimeters. What is the volume of the sphere? Use 3.14 for π and round to the nearest whole number.
7. If a golf ball has a diameter of 4.9 centimeters and a tennis ball has a diameter of 4.8centimeters, find the difference between the volumes of the two balls.
8. Kauri pours the water out of a cylindrical flower vase with a height of 10 inches and a radius of 8 inches into a spherical flower vase. The spherical vase has a radius of 8 inches. Will the water overflow? If so, by how much? If not, how much space is left in the spherical vase?
9. Find the volume of the figure. Use 3.14 for π and round to the nearest whole number

10. Amy is throwing a party and is choosing from the glasses below to serve her punch. Use the information below to answer the questions that follow

• The shape of Glass 1 is a cone with a radius of 10 cm and a height of 16 cm.
• The shape of Glass 2 is a cylinder with a radius of 4 cm and a height of 6 cm.
• The shape of Glass 3 is a hemisphere with a radius of 4 cm with a cylinder on top of it with a radius of 8 cm and a height of 6 cm.

• Amy wants to choose the glass that has the smallest volume so that she doesn’t have to use as much punch. Find the volume of each glass to determine which glass she should choose

Concept Map:

What have we learned:

• Relate volumes of cones and spheres.
• Find the volume of sphere given surface area of the sphere.
• Find the volume of composite figure by adding the volumes of figures in the object.