Radius

Dependencies: Circle, Segment. A derived definition: the fixed length in the definition of a circle, represented by a segment from the center.

Definition

Let $\odot O$ be a circle, and let $P$ be any point on it.

The segment $OP$ from the center to a point on the circle is called a radius. Its length, the distance from $O$ to $P$, is also called the radius and is denoted by $r$.

“Radius” can mean either the segment or its length. The two uses are unambiguous because the definition of a circle guarantees that every such segment has the same length.

Remarks

• Equal radii restate the circle definition. Saying every point on the circle is distance $r$ from the center is exactly saying all radii of one circle are equal.
• The radius determines size. Center fixes position and radius fixes size. Doubling the radius doubles circumference and quadruples area.
• A diameter is twice a radius. A chord through the center is a diameter of length $2r$, the longest chord of the circle.