CC Geometry R
Aim 10: What is the relationship between the ratio of the side lengths of similar
solids and the ratio of their volumes?
Do Now: Reviewing the Scaling principle for area:
(1) For the following pair of similar figures, write the ratio of corresponding side
lengths, in lowest terms. ______
(2) Write the ratio that compares the areas of the similar figures. _______
(3) If the ratio of corresponding side lengths is c : d, then the ratio of the areas is
__________.
Exercise 1: Each pair of solids shown below is similar. Write the ratio of side
lengths a:b comparing one pair of corresponding sides. Then, complete the third
column by writing the ratio that compares volumes of the similar figures. Simplify
ratios when possible.
Similar Figures
Ratio of Side Lengths
Ratio of Volumes
a:b
Volume (A) : Volume (B)
2
3
1
10
5
15
12
36
4
6
12
18
Volume of a general cone:
(1/3)Area of base x height
Similar Figures
Ratio of Side Lengths
a:b
Ratio of Volumes
Volume (A) : Volume (B)
6
4
6
3
4
2
3
4
2
30
20
40
The Scaling principle for VOLUME
If two similar solids S and T are related by a scale factor r, then their respective
3
3
area are related by a factor of r :
r volume (S) = volume (T)
Exercise 2: The solids are similar. Find the indicated measure. a) triangular prisms
20 cm b) cylinders
x
h
5 ft
V = 7π ft3
V = 54 cm3 V = 56π ft3
V = 250 cm3 c) square prisms
9
12 in.
.
in
18
3 in.
d) pentagonal prisms
V= 4608 in3
V = x in3
3
V = 2673 in
V = x in3
Exercise 3:
a. Calculate the volume of this triangular prism:
5
3
3
b. If one side of the triangular base is scaled by a factor of 2, the other side by a
factor of 4, and the height of the prism by a factor of 3, what are the dimensions
of the scaled prism?
c. Calculate the volume of the scaled triangular prism.
d. How does the volume of the scaled prism compare to the volume of the original
prism? Do the volumes have the same relationship we found in exercise 1?
Exercise 4:
a. Calculate the volume of this rectangular prism:
b. If one side of the rectangular base is scaled by a factor of
1/2, the other side by a factor of 24, and the height of the
prism by a factor of 1/3, what are the dimensions of the scaled prism?
c. Calculate the volume of the scaled rectangular prism.
Let's Sum It up!
• If the ratio of the lengths of similar solids is a : b, then the ratio of their
3
3
volumes is a : b .
3
Volume (B) = r x Volume (A)
• When a solid with volume V is scaled by factors r, s, and t in three perpendicular
directions, then the volume of the scaled figure is multiplied by rst.
Volume (T') = rst x Volume (T)
Name ______________________
CC Geometry R
HW #10
Date ________________
1. The heights of two similar prisms are 12cm. and 14 cm. What is the ratio of
their volumes.
2. Two circular cylinders are similar. The ratio of the areas of their bases is 9:4.
Find the ratio of the volumes of the cylinders.
2
2
3. The areas of two similar cylinders are 100 m and 81 m . What is the ratio of
their volumes?
4. The prisms to the right are similar. The volume of
the smaller solid is 1000. Find the volume of the larger.
1
1.1
5. Coffee sold at a deli comes in similar shaped cups. A small cup has a height of
4.2" and a large cup has a height of 5". The large coffee holds 12 fluid oz. How
much coffee is in a small cup? Round your answer to the nearest tenth.
4.2"
5"
V = 12 fl. oz.
6. Right circular cylinder A has volume 2,700 and radius 6. Right circular cylinder
B is similar to cylinder A and has volume 6,400. Find the radius of cylinder B.
8. The volume of a rectangular pyramid is 60. The width of the base is then
scaled by a factor of 3, the length of the base is scaled by a factor of 5/2, and
the height of the pyramid is scaled such that the resulting image has the same
volume 60. Find the scale factor used for the height.
OVER
Review:
1. For each pair of similar figures, write the ratio of side lengths a:b that
compares one pair of corresponding sides. Then write the ratio that compares the
areas of the similar figures. Simplify ratios.
a.
b.
2.
3. The shaded area is composed of a semi-circle minus a hole that is a smaller
semi-circle, and an isosceles triangle minus a hole that is a smaller equilateral
triangle. FD = 5, DB = 3, GD = 13, CB = 17.
A
FF 5
3
E
D
B
13
17
G
C
4. Using a compass and a straightedge, construct circle O' which is the dilation
of circle O using scale factor 2 and A the center of dilation.
A
O
10
1
12.5π