Sean has two circles, one with a radius of 1 inch and another with a radius of 3 inches.

- What is the ratio between the radii of the circles?
- What is the scale factor between the two circles?
- What is the ratio between the circumferences of the circles?
- What is the ratio between the areas of the circles?
- What do area ratios and circumference ratios have to do with scale factor?

### Circles and Similarities

A **circle** is a set of points equidistant from a given point. The **radius** of a circle, \begin{align*}r\end{align*}, is the distance from the center of the circle to the circle. A circle's size depends *only* on its radius.

Two figures are **similar** if a similarity transformation will carry one figure to the other. A **similarity** **transformation** is one or more rigid transformations followed by a dilation. In the examples, you will show that a similarity transformation exists between any two circles and therefore, all circles are similar.

Recall two important formulas related to circles:

- Circumference (Perimeter) of a Circle: \begin{align*}C=2 \pi r\end{align*}
- Area of a Circle: \begin{align*}A=\pi r^2\end{align*}

Once you have shown that all circles are similar, you will explore how the circumferences and areas of circles are related.

#### Proving Circle Similarity

1. Consider circle \begin{align*}A\end{align*}, centered at point \begin{align*}A\end{align*} with radius \begin{align*}r_A\end{align*}, and circle \begin{align*}D\end{align*}, centered at point \begin{align*}D\end{align*} with radius \begin{align*}r_D\end{align*}. Perform a rigid transformation to bring point \begin{align*}A\end{align*} to point \begin{align*}D\end{align*}.

Draw a vector from point \begin{align*}A\end{align*} to point \begin{align*}D\end{align*}. Translate circle \begin{align*}A\end{align*} along the vector to create circle \begin{align*}A^\prime\end{align*}. Note that \begin{align*}r_A \cong r_A^\prime\end{align*}.

2. Dilate circle \begin{align*}A\end{align*} to map it to circle \begin{align*}D\end{align*}. Can you be confident that the circles are similar?

Because the size of a circle is completely determined by its radius, you can use the radii to find the correct scale factor. Dilate circle \begin{align*}A^\prime\end{align*} about point \begin{align*}A^\prime\end{align*} by a scale factor of \begin{align*}\frac{r_D}{r_{A^{\prime}}}\end{align*}.

Circle \begin{align*}A^{\prime \prime}\end{align*} is the same circle as circle \begin{align*}D\end{align*}. You can be confident of this because \begin{align*}r_{A^{\prime \prime}}=\frac{r_D}{r_{A^{\prime}}} \cdot r_{A^{\prime}}=r_D\end{align*} and point \begin{align*}A^{\prime \prime}\end{align*} is the same as point \begin{align*}D\end{align*}. Because a circle is defined by its center and radius, if two circles have the same center and radius then they are the same circle.

This means that circle \begin{align*}A\end{align*} is similar to circle \begin{align*}D\end{align*}, because a similarity transformation (translation then dilation) mapped circle \begin{align*}A\end{align*} to circle \begin{align*}D\end{align*}.

Circle \begin{align*}A\end{align*} and circle \begin{align*}D\end{align*} were two random circles. This proves that in general, all circles are similar.

3. Show that circle \begin{align*}A\end{align*} with center \begin{align*}(-3, 4)\end{align*} and radius 2 is similar to circle \begin{align*}B\end{align*} with center \begin{align*}(3, 2)\end{align*} and radius 4.

Translate circle \begin{align*}A\end{align*} along the vector from \begin{align*}(-3, 4)\end{align*} to \begin{align*}(3, 2)\end{align*}. Then, dilate the image about its center by a scale factor of 2. You will have mapped circle \begin{align*}A\end{align*} to circle \begin{align*}B\end{align*} with a similarity transformation. This means that circle \begin{align*}A\end{align*} is similar to circle \begin{align*}B\end{align*}.

**Examples**

**Example 1**

Earlier, you were given a problem about Sean and his two circles.

Sean has two circles, one with a radius of 1 inch and another with a radius of 3 inches.

a. What is the ratio between the radii of the circles?

The ratio between the radii is \begin{align*}\frac{3}{1}\end{align*}.

b. What is the scale factor between the two circles?

A scale factor exists because any two circles are similar. You can use the radii to determine the scale factor. The ratio between the radii is \begin{align*}\frac{3}{1}\end{align*} so the scale factor is \begin{align*}\frac{3}{1}=3\end{align*}.

c. What is the ratio between the circumferences of the circles?

The circumference of the smaller circle is \begin{align*}C=2 \pi (1)=2 \pi\end{align*}. The circumference of the larger circle is \begin{align*}C=2 \pi (3)=6 \pi\end{align*}. The ratio between the circumferences is \begin{align*}\frac{6 \pi}{2 \pi}=\frac{3}{1}\end{align*}.

d. What is the ratio between the area of the circles?

The area of the smaller circle is \begin{align*}A=\pi(1)^2=\pi\end{align*}. The area of the larger circle is \begin{align*}A=\pi(3)^2=9 \pi\end{align*}. The ratio between the areas is \begin{align*}\frac{9 \pi}{\pi}=\frac{9}{1}\end{align*}. Note that \begin{align*}\frac{9}{1}=\left(\frac{3}{1} \right)^2\end{align*}.

e. What do area ratios and circumference ratios have to do with scale factor?

The area ratio is the scale factor squared, because area is a two dimensional measurement. The circumference ratio is equal to the scale factor, because circumference is a one dimensional measurement.

#### Example 2

Show that circle \begin{align*}A\end{align*} with center \begin{align*}(-1, 7)\end{align*} and radius 2 is similar to circle \begin{align*}B\end{align*} with center \begin{align*}(4, 6)\end{align*} and radius 3.

Translate circle \begin{align*}A\end{align*} along the vector from \begin{align*}(-1, 7)\end{align*} to \begin{align*}(4, 6)\end{align*}. Then, dilate the image about its center with a scale factor of \begin{align*}\frac{3}{2}\end{align*}. You will have mapped circle \begin{align*}A\end{align*} to circle \begin{align*}B\end{align*} with a similarity transformation. This means that circle \begin{align*}A\end{align*} is similar to circle \begin{align*}B\end{align*}.

#### Example 3

The ratio of the circumference of circle \begin{align*}D\end{align*} to the circumference of circle \begin{align*}C\end{align*} is \begin{align*}\frac{4}{3}\end{align*}. What is the ratio of their areas?

The ratio of the circumferences is the same as the scale factor. Therefore, the scale factor is \begin{align*}\frac{4}{3}\end{align*}. The ratio of the areas is the scale factor squared. Therefore, the ratio of the areas is \begin{align*}\left(\frac{4}{3} \right)^2=\frac{16}{9}\end{align*}.

#### Example 4

The ratio of the area of circle \begin{align*}F\end{align*} to the area of circle \begin{align*}E\end{align*} is \begin{align*}\frac{9}{4}\end{align*}. What is the ratio of their radii?

The ratio of the areas is the scale factor squared. Therefore, the scale factor is \begin{align*}\sqrt{\frac{9}{4}}=\frac{3}{2}\end{align*}. The ratio of the radii is the same as the scale factor, so the ratio of the radii is \begin{align*}\frac{3}{2}\end{align*}.

### Review

For #1-#10, show that the circles are similar by describing the similarity transformation necessary to map one circle onto the other.

1. Circle \begin{align*}A\end{align*} with center \begin{align*}(2, 7)\end{align*} and radius 4. Circle \begin{align*}B\end{align*} with center \begin{align*}(1, -4)\end{align*} and radius 3.

2. Circle \begin{align*}A\end{align*} with center \begin{align*}(6, 4)\end{align*} and radius 3. Circle \begin{align*}B\end{align*} with center \begin{align*}(-5, 6)\end{align*} and radius 5.

3. Circle \begin{align*}A\end{align*} with center \begin{align*}(1, 4)\end{align*} and radius 2. Circle \begin{align*}B\end{align*} with center \begin{align*}(3, -2)\end{align*} and radius 7.

4. Circle \begin{align*}A\end{align*} with center \begin{align*}(8, 1)\end{align*} and radius 6. Circle \begin{align*}B\end{align*} with center \begin{align*}(6, -4)\end{align*} and radius 8.

5. Circle \begin{align*}A\end{align*} with center \begin{align*}(-2, 10)\end{align*} and radius 3. Circle \begin{align*}B\end{align*} with center \begin{align*}(-1, -4)\end{align*} and radius 6.

6. Circle \begin{align*}A\end{align*} with center \begin{align*}(-1, 5)\end{align*} and radius 4. Circle \begin{align*}B\end{align*} with center \begin{align*}(-1, 5)\end{align*} and radius 5.

7. Circle \begin{align*}A\end{align*} with center \begin{align*}(-4, -2)\end{align*} and radius 1. Circle \begin{align*}B\end{align*} with center \begin{align*}(1, 8)\end{align*} and radius 4.

8. Circle \begin{align*}A\end{align*} with center \begin{align*}(10, 3)\end{align*} and radius 5. Circle \begin{align*}B\end{align*} with center \begin{align*}(4, 2)\end{align*} and radius 8.

9. Circle \begin{align*}A\end{align*} with center \begin{align*}(12, 4)\end{align*} and radius 10. Circle \begin{align*}B\end{align*} with center \begin{align*}(12, 4)\end{align*} and radius 12.

10. Circle \begin{align*}A\end{align*} with center \begin{align*}(-7, 6)\end{align*} and radius 9. Circle \begin{align*}B\end{align*} with center \begin{align*}(1, -4)\end{align*} and radius 9.

11. The ratio of the circumference of circle \begin{align*}A\end{align*} to the circumference of circle \begin{align*}B\end{align*} is \begin{align*}\frac{2}{3}\end{align*}. What is the ratio of their radii?

12. The ratio of the area of circle \begin{align*}A\end{align*} to the area of circle \begin{align*}B\end{align*} is \begin{align*}\frac{6}{1}\end{align*}. What is the ratio of their radii?

13. The ratio of the radius of circle \begin{align*}A\end{align*} to the radius of circle \begin{align*}B\end{align*} is \begin{align*}\frac{5}{9}\end{align*}. What is the ratio of their areas?

14. The ratio of the area of circle \begin{align*}A\end{align*} to the area of circle \begin{align*}B\end{align*} is \begin{align*}\frac{12}{5}\end{align*}. What is the ratio of their circumferences?

15. To show that any two circles are similar you need to perform a translation and/or a dilation. Why won't you ever need to use reflections or rotations?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.1.