Similar Solids

Key Concepts

• Identify similar solids.
• Use the scale factor of similar solids.
• Find the scale factor.
• Compare similar solids.

Introduction

Similar solids

Two solids with equal ratios of corresponding LINEAR measures, such as heights or radii, are called similar solids. This common ratio is called the scale factor of one solid to the other solid.  

Identify similar solids 

Example 1: 

Compare and decide whether the two solids are similar.  

Solution: 

Lengths = 3/6=1/2

Widths = 2/4=1/2

Heights = 2/2=1/1

The solids are not similar because the ratios of corresponding linear measures are not equal, as shown.  

Example 2: 

Compare and decide whether the two solids are similar.  

Solution: 

Lengths = 3/6=1/2

Widths = 2/4=1/2

Heights = 2/4=1/2

The solids are similar because the ratios of corresponding linear measures are equal, as shown.  

Use the scale factor of similar solids 

Similar Solids Theorem 

If two similar solids have a scale factor of a:b, then corresponding areas have a ratio of a²:b², and corresponding volumes have a ratio of a³:b³.  

The term areas in the theorem can refer to any pair of corresponding areas in similar solids, such as lateral areas, base areas, and surface areas.  

Example 3: 

The 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.  

Solution: 

Begin by using the similar solids theorem to set up the two proportions. 

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. 

Find the scale factor 

Example 4: 

The cubes are similar. Find the scale factor of Cube A to Cube B. 

Solution: 

So, the two cubes have a scale factor of 2:3.  

Compare similar solids 

Example 5: 

Two swimming pools are similar with a scale factor of 3:4. The amount of chlorine mixture to be added is proportional to the volume of water in the pool. If two cups of chlorine mixture are needed for the smaller pool, how much of the chlorine mixture is needed for the larger pool? 

Solution: 

Using the scale factor, the ratio of the volume of the smaller pool to the volume of the larger pool is as follows: 

The ratio of the volumes of the mixture is 1:2.4. The amount of the chlorine mixture for the larger pool can be found by multiplying the amount of the chlorine mixture for the smaller pool by 2.4 as follows: 2(2.4) = 4.8 . 

So, the larger pool needs 4.8 cups of chlorine mixture. 

Exercise

• Two solids with equal ratios of corresponding linear measures are called _______________.
• If two solids are similar, what is the ratio of their surface areas, and what is the ratio of their volumes?
• Determine if both the solids are similar. Explain why or why not.

• Determine if both the solids are similar. Explain why or why not.

• A model train is built with a scale of 1:12. The model train has a surface area of 94 square inches. What is the surface area of the actual train?
• Solid A (shown) is similar to Solid B (not shown) with a scale factor of 1:2. Find the surface area and volume of Solid B.

• Solid A (shown) is similar to Solid B (not shown) with a scale factor of 1:2. Find the surface area and volume of Solid B.

• Solid A (shown) is similar to Solid B (not shown) with a scale factor of 2:3. Find the surface area and volume of Solid B.

• Solid I is similar to Solid II. Find the scale factor of Solid I to Solid II.

• Solid I is similar to Solid II. Find the scale factor of Solid I to Solid II.

Concept Map

What have we learned

• Compare and identify two similar solids.
• Use the scale factor of similar solids and compare two solids using similar solids theorem.
• Find the scale factor of two solids using similar solids theorem.
• Compare two similar solids.