1. Similar Solids and Scale Factor
Learning Target: I can solve word problems using surface area and volume with scale factor.

2. Similar Solids
Two solids of the same type in which their corresponding linear measures (such as heights or radii) form equal ratios. 4 3 6
4.5 3 5 9
5.4

3. Similar Solids: Scale Factor
To compare the ratios of corresponding sides or other linear lengths, write the ratios as fractions in simplest terms. 12 3 6 8 2 4
length: width: height:
Notice that all ratios for corresponding measures are equal in similar solids. The reduced ratio is called the scale factor (a : b).

4. Example: Are these solids similar? Solution:16 12 8 6 9
All corresponding ratios are equal, so the figures are similar.

5. Are these solids similar?
Example:
Are these solids similar? 8 18 4 6
Solution:
Corresponding ratios are not equal, so the figures are not similar.

6. Similar Solids: Area & Volume Ratios
If two similar solids have a scale factor of a : b,
then the ratio of their areas is a2 : b2
then the ratio of their volumes is a3 : b3 4 3 6
4.5
scale factor
area ratio
volume ratio

7. Ex. 1: Using the scale factor of similar solids as a proportion
Two prisms are similar with a scale factor of 1:3.
Find the surface area and volume of prism G given that:
the surface area of prism F is 24 square feet and
the volume of prism F is 7 cubic feet.

8. Solution:
Begin by using what we know about scale factor to set up a proportion
Surface area of F
Surface area of G = a2 b2
Volume of F
Volume of G = a3 b3 24
Surface area of G = 12
32 = 9 7
Volume of G = 13
33 = 27
Surface area of G = 216
Volume of G = 189
So, the surface area of prism G is 216 square feet and the volume of prism G is 189 cubic feet.

9. Check your work You can check your work by
substituting back into original
proportions.
Surface area of F
Surface area of G = a2 b2
Volume of F
Volume of G = a3 b3
Surface area of F
Surface area of G = 24
216 1 9
Volume of F
Volume of G =
Surface area of G = 216
Volume of G = 189
So, the surface area of prism G is 216 square feet and the volume of prism G is 189 cubic feet.

10. Ex. 2: Finding the scale factor of similar solids
To find the scale factor of the two cubes, find the ratio of the two volumes. a3 b3
512
1728 =
Write ratio of volumes. a b 8 12 =
Use a calculator to take the cube root. = 2 3
Simplify.
So, the two cubes have a scale factor of 2:3.

11. Word Problem Examples
Below is a 1:8 scale model of the Lamborghini Aventador, built using the same materials as the real thing.
If it costs $135 to paint the model, and $725 to build all the parts, how much would the real thing cost?

12. Exit Ticket
The two cylinders are similar with the given scale factor. Find the surface area S and volume V of the smaller solid.
9 cm
15 cm
10 cm
Surface Area of Smaller Cylinder

13. Exit Ticket
The two cylinders are similar with the given scale factor. Find the surface area S and volume V of the smaller solid.
9 cm
15 cm
10 cm
Volume of Smaller Cylinder

14. Exit Ticket
Use the given information about the two similar solids to find their scale factor.
Scale Factor