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

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Recall the formula for the length of an arc intercepted by a central angle: $L = r \theta$, where $L$ is the arc length, $r$ is the radius of the circle, and $\theta$ is the central angle in radians.

Note that the central angle $\theta$ is fixed, so it remains constant in this problem.

If the radius $r$ is doubled, then the new radius becomes \$2r$.

Substitute the new radius into the arc length formula to find the new arc length: $L_{new} = (2r) \theta$.

Compare the new arc length $L_{new}$ to the original arc length $L$ to determine how the arc length changes when the radius is doubled.

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Arc Length Formula

The arc length of a circle is calculated using the formula L = rθ, where 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.

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Finding Missing Side Lengths

Effect of Radius on Arc Length

Since arc length depends linearly on the radius, doubling the radius while keeping the central angle fixed will double the arc length. This means the arc length changes proportionally with the radius.

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Finding Missing Side Lengths

Central Angle in Radians

The central angle must be measured in radians for the arc length formula L = rθ to be valid. Radians relate the angle directly to the radius, making calculations of arc length straightforward and consistent.

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