Circle Properties

Understanding Circles

A circle is a set of points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius.

Key terms related to circles include:

• Radius: A line segment from the center to any point on the circle.
• Diameter: A line segment that passes through the center and has its endpoints on the circle. It is twice the radius.
• Circumference: The distance around the circle.
• Arc: A part of the circumference of a circle.
• Sector: A region bounded by two radii and an arc.

Properties of Exterior Angles

An exterior angle of a circle is formed when a tangent and a chord intersect at a point on the circle. The measure of an exterior angle is equal to the difference between the measures of the intercepted arcs.

For example, if the exterior angle is formed by a tangent and a chord, and the intercepted arcs are $a$ and $b$, then:

$$\text{Exterior Angle} = \frac{1}{2} (a - b)$$

Properties of Similar and Equiangular Circles

Circles are similar if they have the same shape but different sizes. All circles are similar because they can be transformed into one another by scaling.

Circles are equiangular if they have the same angle measures. In circles, all angles subtended by the same arc are equal.

Worked Example

Examples

Example 1: Similar Circles

If two circles have radii of 3 cm and 6 cm, they are similar because they have the same shape and their radii are proportional.

Example 2: Equiangular Circles

In two circles, if an arc subtends an angle of 60° at the center of both circles, the circles are equiangular.