Concept Check If the radius of a circle is doubled, how is the length of the arc intercepted by a fixed central angle changed?

Key Concepts

Arc Length Formula

The length of an arc in a circle can be calculated using the formula L = rθ, where L is the arc length, r is the radius, and θ is the central angle in radians. This formula shows that the arc length is directly proportional to both the radius and the angle, meaning that if the radius increases, the arc length will also increase proportionally.

Central Angle

A central angle is an angle whose vertex is at the center of the circle and whose sides intersect the circle. The measure of the arc length intercepted by this angle depends on the size of the angle and the radius of the circle. When the radius is doubled, the arc length for the same central angle will also change, illustrating the relationship between the angle and the radius.

Proportional Relationships

In geometry, proportional relationships indicate how two quantities change in relation to each other. In the context of the arc length and radius, if the radius is doubled, the arc length will also double for the same central angle. Understanding this proportionality is crucial for predicting how changes in one dimension affect another in circular geometry.