CC Geometry H
Aim #9: What is the relationship between the ratio of the side lengths of similar
solids and the ratio of their volumes?
Do Now: For the following pair of similar figures, write the ratio of side lengths
c:d that compares one pair of corresponding sides. Then write the ratio that
compares the areas of the similar figures. Simplify ratios.
When the ratio of side lengths is c : d, then the ratio of the areas is __________.
Make a conjecture as to how the ratio of sides c : d will be related to the ratio of
volumes Volume (C) : Volume (D).
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
a:b
Ratio of Volumes
Volume (A) : Volume (B)
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)
36
18
1) b. Suppose a similarity transformation takes a solid S to a solid T at scale
factor r. How do you think the volume of S compares to the volume of T?
2) 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.
3) Right circular cylinder A has volume 2,700 and radius 3. Right circular cylinder
B is similar to cylinder A and has volume 6,400. Find the radius of cylinder B.
4) a. Calculate the volume of this triangular prism:
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. Make a conjecture about the relationship between the volume of the original
prism and the scaled prism. Do the volumes have the same relationship we found in
problem 1?
5) 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.
d. Make a conjecture about the relationship between the volume of the original
prism and the scaled prism.
e. If a solid T is scaled by factors r, s, and t in three perpendicular directions, what
happens to the volume?
Let's Sum It up!
-If the ratio of the lengths of similar solids is a : b, then the ratio of their volumes
3
3
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 ______________________
Date ________________
CC Geometry H
HW #9
1. Two circular cylinders are similar. The ratio of the areas of their bases is 9:4.
Find the ratio of the volumes of the similar solids.
2. 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
as the original pyramid. Find the scale factor used for the height.
3. The following solids are similar. The volume of the 1st solid is 1000. Find the
volume of the second.
4. If the sides of two similar triangles are in the ratio of 2:3, find the ratio of
their perimeters, areas, and volumes. 5. The ratio of the sides of two similar cubes is 3:4. the smaller cube has a volume
3
of 729 m . What is the volume of the larger cube?
6. The Empire State Building is a 102 story skyscraper. Its height is 1,250 ft from
the ground to the roof. The length and width of the building are approximately
424 ft and 187 ft, respectively. A company plans to make a miniature version of
the building and sell cases to souvenir shops.
a. The miniature version is just 1/2500 of the size of the original. What are the
dimensions of the miniature Empire State Building? Round to the nearest
hundredth when necessary.
b. Determine the volume of the miniature building. Let us assume that the building
is in the shape of a rectangular prism.
Review:
1. For each pair of similar figures, write the ratio of the side lengths a:b. Then
write the ratio that compares the areas of the similar figures. Simplify ratios.
a.
2.
b.