Find the length of the arc on a circle of radius r...

Find the length of the arc on a circle of radius $r$ intercepted by a central angle $\boldsymbol{\theta}$
Central Angle $\boldsymbol{\theta}$
$180^{\circ}$
$60^{\circ}$
$\frac{2 \pi}{3}$ radians
$\frac{3 \pi}{4}$ radians
Radius $r$
14 inches

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Key Concepts

Arc Length

Arc length is the distance along a section of the circumference of a circle. It is calculated by multiplying the radius of the circle by the central angle (in radians) that intercepts the arc. This formula connects the linear distance of the arc with the angular measure, emphasizing the direct proportionality between them.

Central Angle

A central angle is an angle whose vertex is at the center of the circle and whose sides extend to intersect the circle. It plays a crucial role in determining the portion of the circle’s circumference (arc) that is being considered, as measured by the arc length formula.

Radians

Radians are a unit of angular measure based on the radius of a circle. One radian is defined as the angle created when the arc length equals the radius of the circle. Using radians simplifies calculations in many areas of mathematics and physics, particularly when relating angles to arc lengths.

Conversion Between Degrees and Radians

The conversion between degrees and radians is essential when working with angles that are not initially given in a single unit. Specifically, the relationship ? radians = 180° allows for the conversion of angles from degrees to radians, which is necessary when using formulas that require radian measure, such as the arc length formula.

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