Circle geometry

The geometry in this chapter is important to know well. Many advanced topics in geometry and coordinate geometry (especially problems involving circles) build directly on these principles.

Core principles of circle geometry

Radius

The radius ($r$) of a circle is the distance from the center of the circle to the edge of the circle. Because a circle is perfectly round, the radius has the same length no matter which point on the edge you choose.

Diameter

The diameter ($d$) of a circle is the distance from one point on the edge of the circle, through the center, to the opposite edge of the circle. The diameter must pass through the center. That’s what makes it the longest straight-line distance across the circle.

The diameter is equal to twice the radius:

$$d=2\ast r$$

Area

To find the area of a circle, you need the radius. In fact, the radius is all you need.

The area of a circle is:

$$A=\pi r^2$$

Where $A$ is the area, $r$ is the radius, and $\pi$ (pi) is a constant equal to about $3.14$. Most calculators have a $\pi$ button.

On tests, you may be asked to leave your answer in terms of $\pi$ instead of using a decimal approximation. For example, if $r=2$:

$$\begin{array}{rl}A& =4\pi \\ A& \approx 12.57\end{array}$$

Circumference

The circumference of a circle is the distance around the outside of the circle. You can think of it as the circle’s perimeter.

Circumference can also be found using just the radius:

$$C=2\pi r$$

In this equation, $C$ is the circumference and $r$ is the radius.

Arc length

Arc length measures the length of part of the circumference. In equations, arc length is represented by $s$. The arc length depends on the central angle $\theta$ (theta).

To find the arc length for an angle $\theta$, use one of these formulas depending on the units.

With $\theta$ in degrees (not radians):

$$s=2\pi r(\theta /360)$$

You can extend this idea to radians by remembering that a full circle is $2\pi$ radians.

With $\theta$ in radians (not degrees):

$$s=2\pi r(\theta /(2\pi ))$$

This simplifies to:

$$s=r\theta$$

Angle measurements

A full circle measures 360 degrees (or $2\pi$ radians). That means the angles around the center of a circle add up to 360 degrees. If you know all but one of the angles, you can find the missing angle by subtracting the known angles from 360 degrees.

Diagram

Below is an image showing the core principles of circle geometry:

Examples:

Find the circumference of a circle that has a radius of $3$.

Here’s another:

If a circle has a diameter of $8$, what is its area?

Let’s do one more focused on angles:

What is the measure of angle $X$?