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Chapter 9—Basic Geometry

9-4 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 = in3 cubic meters = m3 cubic miles = mi3

Volume of Rectangular Solids and Cubes

A rectangular solid is a 3-dimensional 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 solid--one 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 = s3 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 = s3 Assign the given values V = 103 Expand the expression V = 1000

The volume of this cube is 1000 in3.

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 = 1/3b2h 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 = 1/3b2h Assign the given values V = 1/3 · 22 · 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 = pr2h 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 = pr2h Assign the given values V = 3.14 · 2.52 · 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 pr3 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 = 4/3pr3 Assign the given values V = 4/3 · 3.14 · 103 Complete the solution V = 4186.667Rounding to the nearest 1s unit:      V = 4187 Solution V = 4187 ft3The 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 = 1/3pr2h 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: