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Chapter 9—Basic Geometry
94 Finding Volumes
When you complete the work for this section, you should be able to:  Explain the meaning of volume.
 Given the dimensions of a rectangular solid, cube, pyramid, cylinder, or cone, calculate its volume.
 Given the diameter or radius of a sphere, calculate its volume.

Six solid figures of basic geometry.
Definition The volume of a geometric solid is the number of cubes of a given size that fill the space of the object. 
The volume of a solid object is a measure of the amount of space is occupies. Generally speaking, the larger the dimensions of a solid object, the larger its volume.
Volume is expressed in cubic units such as cubic inches, cubic feet, cubic miles, and so on. Units of volume can also be abbreviated by using the cube symbol. For example::
 cubic inches = in^{3}
 cubic meters = m^{3}
 cubic miles = mi^{3}
 
Volume of Rectangular Solids and Cubes
A rectangular solid is a 3dimensional version of a plane rectangle.
Here are a few more facts to consider about rectangular solids:
 They have six rectangular surfaces.
 All angles are right angles.
 Parallel edges have the same length.
 It has 8 corners and 12 edges.
Equation  The equation for the volume of any rectangular solid is: V = lwh Where:  l is the length
 h is the height
 w is the width
 V is the volume of the figure
 
Example
The Problem
Determine the volume of a rectangular solid that is 10 inches long, 15 inches wide, and 8 inches high.
The Solution
Cite the appropriate equation  V = lwh 
Assign the given values  V = 10 · 15 · 8 
Complete the solution  V = 1200 
This rectangular solid has a volume of 1200 cubic inches.
Examples and Exercises
Use these interactive examples and exercises to strengthen your understanding and build your skills:  Note: Use this figure only for reference. The proportions do not necessarily match those cited in the problem.  
A cube is a special kind of rectangular solidone where all sides have the same length.
Here are a few more features to consider:
 A cube has six square surfaces.
 All angles are right angles.
 It has 8 corners and 12 edges.
Equation  The equation for the volume of any cube: V = s^{3} Where:  s is the length of the sides
 V is the volume of the cube
 
Example
The Problem
One side of a cube measures 10 in. What is the volume of this cube?
The Solution
Cite the appropriate equation  V = s^{3} 
Assign the given values  V = 10^{3} 
Expand the expression  V = 1000 
The volume of this cube is 1000 in^{3}.
Examples and Exercises
Use these interactive examples and exercises to strengthen your understanding and build your skills:   
Volume of a Pyramid
The pyramid used in this lesson has these characteristics:
 It has a square base.
 It includes 4 identical triangles that meet at the apex of the pyramid.
Equation  The equation for the volume of a pyramid: V = ^{1}/_{3}b^{2}h Where:  b is the length of the bases
 h is the height
 V is the volume of the pyramid
 
Example
The Problem
Determine the volume of a pyramid that has a base of 2 m and a height of 6 m.
The Solution
Cite the appropriate equation  V = ^{1}/_{3}b^{2}h 
Assign the given values  V = ^{1}/_{3} · 2^{2} · 6 
Complete the solution  V = 8 
The volume of this pyramid is 8 cubic meters.
Examples and Exercises
Use these interactive examples and exercises to strengthen your understanding and build your skills:  Note: Use this figure only for reference. The proportions do not necessarily match those cited in the problem.  
Volume of a Cylinder
Equation  The equation for the volume of a cylinder: V = pr^{2}h Where:  p = approximately 3.14 or ^{22}/_{7}
 r is the radius
 h is the height of the cylinder
 V is the volume of the cylinder
 
Example
The Problem
A certain cylinder has a radius of 2.5 inches and a height of 8 inches. What is the volume of this cylinder?
The Solution
Cite the appropriate equation  V = pr^{2}h 
Assign the given values  V = 3.14 · 2.5^{2} · 8 
Complete the solution  V = 157 
The volume of this cylinder is 157 cubic inches.
Examples and Exercises
Use these interactive examples and exercises to strengthen your understanding and build your skills:   
Volume of a Sphere
Equation  The equation for the volume of a sphere: V = ^{4}/_{3 }pr^{3} Where:  p = approximately 3.14 or ^{22}/_{7}
 r is the radius
 V is the volume
 
Example
Determine the volume of a sphere that has a radius of 10 ft.
Problem   Find the volume of a sphere where r = 10 ft. 
Cite the appropriate equation  V = ^{4}/_{3}pr^{3}  
Assign the given values  V = ^{4}/_{3} · 3.14 · 10^{3}  
Complete the solution  V = 4186.667 Rounding to the nearest 1s unit: V = 4187  
Solution   V = 4187 ft^{3} The volume is slightly less than 4187 cubic feet 
Examples and Exercises
Use these interactive examples and exercises to strengthen your understanding and build your skills:   
Volume of a Cone
Equation  The equation for the volume of a cone: V = ^{1/}_{3}pr^{2}h Where:  p = approximately 3.14 or ^{22}/_{7}
 r is the radius
 h is the height
 V is the volume
 
Examples
Examples and Exercises
Use these interactive examples and exercises to strengthen your understanding and build your skills:   