Lesson 1: Circles and Circumference

Property

A circle is the set of all points in a plane that lie at a given distance from a fixed point. The fixed point is called the center of the circle, and the distance from the center to any point on the circle is called the radius.

Examples

Property

• $r$ is the length of the radius
• $d$ is the length of the diameter
• $d = 2r$

Remember that we approximate $\pi$ with $3.14$ or $\frac{22}{7}$ depending on whether the radius of the circle is given as a decimal or a fraction.

Examples

• A circle has a radius ($r$) of 5 cm. Its diameter ($d$) is calculated as $d = 2r = 2(5) = 10$ cm.

• If a circle's diameter is 14 inches, its radius is half of that. We find the radius by solving $14 = 2r$, so $r = 7$ inches.

Property

The distance from the center of a circle to any point on the circle itself is called the radius of the circle.
The diameter of a circle is the length of a line segment joining two points on the circle and passing through the center. Thus, the diameter of a circle is twice its radius.
The perimeter of a circle is called its circumference.
The circumference $C$ of a circle is given by

where $d$ is the diameter of the circle. The Greek letter $\pi$ (pi) stands for an irrational number: $\pi = 3.141592654 ...$

Examples

• A circular pool has a diameter of 10 meters. Its circumference is $C = \pi \times 10 \approx 31.42$ meters.
• A bicycle wheel has a radius of 14 inches. Its diameter is $2 \times 14 = 28$ inches, so its circumference is $C = \pi \times 28 \approx 87.96$ inches.
• If a running track has a circumference of 400 meters, its diameter can be found by $d = \frac{C}{\pi} = \frac{400}{\pi} \approx 127.32$ meters.

Explanation

Circumference is the special name for a circle's perimeter. It's the distance around the circle's edge. This distance is always a little more than 3 times the circle's diameter, a constant ratio we call pi ($\pi$).