## 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 a^{2}:b^{2}, and corresponding volumes have a ratio of a^{3}:b^{3}.

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.

#### Related topics

#### Addition and Multiplication Using Counters & Bar-Diagrams

Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […]

Read More >>#### Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […]

Read More >>#### How to Write and Interpret Numerical Expressions?

Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division A → Addition S → Subtraction Some examples […]

Read More >>#### System of Linear Inequalities and Equations

Introduction: Systems of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this, y < x + 2…………..Inequality 1 y ≥ 2x − 1…………Inequality 2 Solution of a System of Linear Inequalities: A solution of a system of linear inequalities […]

Read More >>
Comments: