ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 26: VOLUMES

SECTION 26.1 THE CONCEPT OF VOLUME

The concept of "volume" applies to objects that are "solid bodies." The "volume" of a "solid body" is the amount of "space" it occupies. Below are some examples of "solid bodies."

SECTION 26.2 WHAT IS A CUBE?

First you must understand the concept of the solid called a "cube." A cube looks like the following:

A cube has six "faces," each face is a square and all six squares are the same size, i.e. congruent.

A cube has 12 "edges" which are the sides of the square faces. All edges are the same length. Each edge meets another edge at a 90 angle.

SECTION 26.3 HOW IS VOLUME MEASURED AND WHAT IS A UNIT CUBE?

You are familiar with the following concepts of measurements called "units": inch foot, yard, mile, etc. I talked about units in a earlier section. I will start with the most fundamental volume of a solid - "the unit cube or 1 cubic unit."

A cube whose edges are all 1 inch is said to have a volume of 1 cubic inch. See image below:

A cube whose edges are all 1 foot is said to have a volume of 1 cubic foot. See image below:

A cube whose edges are all 1 yard is said to have a volume of 1 cubic yard. See image below:

Now I can continue this process with any form of measurement. For example a volume of 1 mi., or 1 cubic mile, is a cube whose faces sides are 1 mile on each edge.

So now that you understand that the basic definition of volume is based on what is called a "unit cube," which is "a cube whose edges all measure one unit" whether that unit be inches, feet, millimeters, yard, kilometers, etc.

SECTION 26.4 EXTENDING THE CONCEPT OF VOLUME?

So what would 10 square feet ( 10 ft.) mean? Well, it means that you have a solid that would contain 10 unit cubes.

What would 30 cubic miles(30 mi.) be? It means that you have a solid that would contain 30 unit cubes.

Hopefully you can appreciate the size of the volume of the earth which is 1,097,509,500,000,000,000,000 cubic meters.

SECTION 26.5 IMPORTANT VOLUME FORMULAS

The following volume formulas are important to know:

VOLUME = length x width x height = width ³ = height ³ = length ³

Please note that in a cube the length, width and height are all equal.

VOLUME = length x width x height units³

VOLUME = (4/3)πr³ units³
where r is the radius of the sphere

VOLUME = (1/3)πr²h units³
where r is the radius of the base and h is the height of the cone.

VOLUME = πr²h units³
where r is the radius of the base of the cone and h is the height of the cone

Examples:

Find the volumes of the following:

a) A sphere with radius 10 feet.

V = (4/3)π(10)³ = 4188.790 in.³

Explanation: There are 4188.790 cubes with each edge of 1 inch that can be packed into this sphere with radius of 10 feet.

b) A rectangular solid with length 30 yards, width 20 yards, height 50 yards.

V = (30)(20)(50)yd.³ = 30000 yd.³

Explanation: There are 30000 cubes with each edge of 1 yard that can be packed into this rectangular solid.

c) A cube with each side 5 miles

V = (5)(5)(5) mi.³ = 125 mi.³

Explanation: There are 125 cubes with each edge of 1 mile that can be packed into this cube.

d) A cone with a base radius of 600 inches and a height of 3200 inches

V = (1/3)π(600)²(3200) in.³ = 1,206,371,579 in.³

Explanation: There are 1, 206, 371, 579 cubes with each edge of 1 inch that can be packed into this cone.

e) A cylinder with base radius of 123 kilometers and a height of 345 kilometers.

V =π(123)² (345)km.³= 16397558.56 km.³

Explanation: There are 16397558.56 cubes with each edge of 1 kilometer that can be packed into this cylinder.