Section 2.3 Volume
Volume is the amount of 3-dimensional space an object occupies.
Consider the diagram below.
For this example the volume is 4cm×3cm×5cm = 60 cm3
This figure is composed of 60 cubes that are each 1 cm by 1 cm by 1 cm. They each have a
volume of 1 cubic centimeter
Here are the volume formulas we will need for this section.
Volume of a “triangular prism”.
Volume of a rectangular pyramid.
Example: Find the volume.
Step 1: Identify the name of the shape and the formula to use.
This is a rectangular pyramid.
1
Formula: 𝑉 = 3 𝐵ℎ 𝑤ℎ𝑒𝑟𝑒 𝐵 = 𝑙𝑤
We can rewrite the volume formula as follows:
1
𝑉 = 3 𝑙𝑤ℎ
Step 2: Assign values to each variable and plug the numbers in the formula.
l = 8 in, w = 7 in h = 6 in (the values of “l” and “w” are interchangeable, although we usually
call the longer side in a rectangle the length.)
1
𝑉 = (8𝑖𝑛)(7𝑖𝑛)(6𝑖𝑛)
3
Answer: Volume = 112 in3 (my units are cubic inches)
Example: Find the volume.
Step 1: Identify the name of the shape and the formula to use.
This is a triangular prism.
1
Formula: 𝑉 = 𝑏ℎ𝑙
2
Step 2: Assign values to each of the variables.
b = 15 cm, h = 8 cm, l = 12 cm
1
𝑉 = 2 (15𝑐𝑚)(8𝑐𝑚)(12𝑐𝑚)
Answer: Volume 720 cm3
Example: Find the volume.
Step 1: Identify the name of the shape and the formula to use.
This is a right circular cone.
1
Formula: 𝑉 = 3 𝜋𝑟 2 ℎ
Step 2: Assign values to each variable and plug the numbers in the formula.
r = 5.7 cm, h = 12 cm
1
𝑉 = 3 𝜋(5.7 𝑐𝑚)2 (12𝑐𝑚)
Answer: Volume = 22.8𝜋𝑐𝑚3
Example: Find the volume.
Step 1: Identify the shape.
This is a combination of two shapes. The top is one half of a sphere. The bottom is a right
circular cone.
Formulas:
1
Volume of right circular cone: 𝑉 = 3 𝜋𝑟 2 ℎ
4
Volume of a sphere 𝑉 = 3 𝜋𝑟 3
Volume of half of a sphere
1 4
𝑉 = ∗ 𝜋𝑟 3 (I need to multiply the volume by ½) (Cancel the 4 and 2)
2
3
2
Volume of half a sphere = 𝑉 = 3 𝜋𝑟 3
Step 2: Assign values and plug numbers in the formula.
Cone r = 4 cm, h = 10 cm
Sphere r = 4 cm
Cone part
+ Sphere part
1
2
2 (10𝑐𝑚)
𝑉 = 3 𝜋(4𝑐𝑚)
+ (3 𝜋(4𝑐𝑚)3 )
1
2
3
3
V = 𝜋(160𝑐𝑚)3 + ( 𝜋64𝑐𝑚3 )
V=
V=
160
3
288
3
𝜋𝑐𝑚3 +
128
3
𝜋𝑐𝑚3
𝜋𝑐𝑚3
Answer: Volume = 96𝜋𝑐𝑚3
Homework # 1- 30: Find the volume. Leave your answer in terms of 𝜋 when appropriate.
1)
2)
3)
4) Assume measurements are in feet.
5)
6)
7)
8)
9)
10)
11) Assume measurements are in
centimeters.
12)
13) Assume the base of the pyramid is a
square. Leave your answer in fraction form.
14) Assume that the units are given in meters.
Round your answer to 2 decimals, if
appropriate.
15) This is a square pyramid. The base is a 4
cm by 4 cm square.
16)
17)
18)
19) Write your answer as a fraction in terms
of 𝜋.
20) Write your answer as a fraction in terms
of 𝜋.
21) Write your answer as a fraction in terms
of 𝜋.
22) Write your answer as a fraction in terms
of 𝜋.
23) Hint you will need to subtract to find the
height of the cone. Write your answer as a
fraction in terms of 𝜋.
24) Write your answer as a fraction in terms of
𝜋.
25)
Write your answer in terms of 𝜋.
26)
𝜋.
Write your answer as a fraction in terms of
27)
28) Assume lengths are given in feet.
29)
30) Assume lengths are given in inches.
31)
32)
Answers: 1) 194.4 cm3 3) 36 ft3 5) 192𝜋 cm3 7) 4.608𝜋 ft3 9) 96𝜋 cm3 11) 960𝜋 cm3
175
10976
32
13) 3 m3 15) 48 cm3 17) 112 in3 19) 3 𝜋 m3 21) 3 𝜋 in3 23) 60840𝜋 ft3
25)
475
3
𝜋 cm3 27) 24000 ft3 29) 1428 in3 31) 24000 mm3