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 units**^{3}
**VOLUME = (4/3)πr**^{3} units^{3}
where r is the radius of the sphere
**VOLUME = (1/3)πr**^{2}h units^{3}
where r is the radius of the base and h is the height of the cone.
**VOLUME = πr**^{2}h units^{3}
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. |