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# Circle

Although you may be familiar with the shape of a circle, it is helpful to define it mathematically. A circle is defined as the set of all points in a plane equidistant from a given point, known as the center. The distance from the center to any point on the circumference is referred to as the radius. A line segment connecting two points on the circumference of the circle and passing through its center is known as the diameter of the circle. The following illustration provides a visual representation of the radius and diameter:

The radius is equal to half of the diameter. This mathematical relationship can be expressed as $d=2r$ , where r represents the radius and d represents the length of the diameter.

## Calculating the circumference of circles

The circumference of a circle is the distance around the outside. You can use the following equation to calculate it for any circle:

$C=2\pi \left(r\right)$

In this equation, C represents the circumference, r is the circle's radius, and the π symbol represents pi: an irrational number that is approximately 3.14. For instance, let's say we have a circle with a radius of 4 meters. We would calculate the circumference as follows:

$C=2\pi \left(4\right)$

$C=8\pi$

C = approximately 25.12 meters (using 3.14 for π)

## Calculating the area of circles

You can calculate the area of any circle using the following formula:

$A=\pi {r}^{2}$

In this formula, A represents the area, π is pi, and r is the radius. If we return to the example above (the circle with a radius of 4 meters), calculating the area would look like this:

$A=\pi {\left(4\right)}^{2}$

$A=16\pi$

A = approximately 50.24 sq. meters (using 3.14 for π)

## Proving all circles are similar

It can be shown that any two circles in the plane are similar as follows:

If there are two circles (Circle A centered at $\left({h}_{1},{k}_{1}\right)$ with radius ${r}_{1}$ and Circle B centered at $\left({h}_{2},{k}_{2}\right)$ with radius ${r}_{2}$ ) we can translate Circle A ${h}_{2}-{h}_{1}$ units to the right and ${k}_{2}-{k}_{1}$ units up so that it's now centered on top of circle B. Note that we might actually be moving the circle left and/or down if ${h}_{2}-{h}_{1}$ and/or ${k}_{2}-{k}_{1}$ is negative.

Next, we can perform a dilation centered at $\left({h}_{2},{k}_{2}\right)$ by a scale factor of $\frac{{r}_{2}}{{r}_{1}}$ . The result is a circle centered at $\left({h}_{2},{k}_{2}\right)$ with a radius of ${r}_{2}$ . Since we were able to transform Circle A into Circle B using nothing but translation and dilation, the two figures must be similar.

## Circles practice questions

a. What is the circumference of a circle with a radius of six inches? Use 3.14 for pi.

$C=2\pi r$

$C=2\left(3.14\right)\left(6\right)$

$37.68\mathrm{inches}$

b. What is the area of a circle with a radius of six inches? Use 3.14 for pi.

$A=\pi {r}^{2}$

$A=\left(3.14\right)\left(36\right)$

c. What is the circumference of a circle with a diameter of 10 feet? Use 3.14 for pi.

$C=\pi \left(d\right)$

$C=\left(3.14\right)\left(10\right)$

d. What is the area of a circle with a diameter of 10 feet? Use 3.14 for pi.

$A=\pi {r}^{2}$

$r=\frac{d}{2}$

$A=\left(3.14\right){\left(5\right)}^{2}$

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