GEOMETRY
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
Name________________________
M3
Period: _____Date______________
Lesson 7: Scaling Factor and Applications of Volumes
Learning Targets
1. I can apply the formula of volume to solve world problems.
2. I can identify the scaling factor in similar solid figures and use that to solve problems
Opening Activity. A cone fits inside a cylinder so that their bases are the same and their heights are the same, as shown
in the diagram below. Calculate the volume that is inside the cylinder but outside of the cone. Give an exact answer.
Scaling Principle for Volumes
Similar Figures ( Areas)
Scale factor = Ratio of
Sides
𝒂: 𝒃 or 𝒄: 𝒅
Ratio of Areas
𝐀𝐫𝐞𝐚(𝑨): 𝐀𝐫𝐞𝐚(𝑩)
△𝑨~△𝑩
Rectangle 𝑨 ~ Rectangle 𝑩
Conclusion: When the ratio of side lengths is a: b, then the ratio of the areas is a2 : b2 .
GEOMETRY
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
Name________________________
M3
Period: _____Date______________
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.
Ratio of Side Lengths
𝒂: 𝒃
Similar Figures
Ratio of Volumes
𝐕𝐨𝐥𝐮𝐦𝐞(𝑨): 𝐕𝐨𝐥𝐮𝐦𝐞(𝑩)
Figure B
Figure A
Figure B
Figure A
Conclusion: For two similar figures whose corresponding lengths are in the ratio 𝑎: 𝑏, the ratio of their volumes is 𝑎3 : 𝑏 3 .
Example 1. Fill in the table
Shape
Scale factor
Scale factor
Ratio of sides - ( r )
cone
cylinder
Prism
𝒓=
Ratio of surface areas
( r2 )
Scale factor
Ratio of volumes ( r3 )
𝟐
𝟓
𝟏𝟔
𝟐𝟓
𝟏𝟐𝟓
𝟔𝟒
GEOMETRY
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
Name________________________
M3
Period: _____Date______________
Example 2. The cones below are similar. What is the volume of the larger cone?
Example 3. 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.
Quick Check for Understanding
1. The following solids are similar. The volume of the first solid is 𝟏𝟎𝟎. Find the volume of
the second solid.
2. Coffee is sold 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 ounces. How much coffee is in a small cup? Round your answer to the
nearest tenth of an ounce.
GEOMETRY
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
Name________________________
M3
Period: _____Date______________
Lesson 7: Scaling Factor and Applications of Volumes
Problem set
1. Fill in the table
Shape
Scale factor
Scale factor
Ratio of sides - ( r )
Ratio of surface areas
( r2 )
Ratio of volumes ( r3 )
𝟐𝟏𝟔
𝟏𝟐𝟓
cone
𝟗
𝟏𝟔
cylinder
Prism
Scale factor
𝒘=
𝟐
𝟑
2. Two right prisms have similar bases. The first prism has height 𝟓 and volume 𝟏𝟎𝟎. A side on the
base of the first prism has length 𝟐, and the corresponding side on the base of the second prism
has length 𝟑. If the height of the second prism is 𝟔, what is its volume?
3. Solid A is similar to Solid B . Find the scale factor of solid A to Solid B . Than find the ratio of the
volumes
4. The area of the base of a cone is 81 cm2 and the cross sectional area cut 4 cm from the vertex of the cone
has an area of 25 cm2. Find the height.