Circular Arcs Find the length s of the circular arc, radius...

Circular Arcs Find the length $s$ of the circular arc, radius $r$ of the circle, or the central angle $\theta,$ as indicated. A central angle $\theta$ in a circle of radius 9 $\mathrm{m}$ is subtended by an arc of length 14 $\mathrm{m} .$ Find the measure of $\theta$ in degrees and radians.

Circular Arcs Find the length $s$ of the circular arc, radius $r$ of the circle, or the central angle $\theta,$ as indicated.
A central angle $\theta$ in a circle of radius 9 $\mathrm{m}$ is subtended by an arc of length 14 $\mathrm{m} .$ Find the measure of $\theta$ in degrees and radians.

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Key Concepts

Arc Length Formula

In circle geometry, the length of an arc is directly proportional to the radius of the circle and the measure of the central angle (in radians) that intercepts the arc. This relationship is expressed by the formula s = r?, where s is the arc length, r is the radius, and ? is the central angle in radians. This formula is fundamental when solving problems involving parts of a circle.

Radians and Degrees

Radians and degrees are two units of measuring angles. One complete revolution (360 degrees) is equivalent to 2? radians. Converting between these units is essential since many formulas in circle geometry, such as the arc length formula, require the angle to be in radians, while practical problems or real-world applications might present angles in degrees.

Central Angle

A central angle is an angle whose vertex is at the center of the circle and whose sides extend to the circumference. It plays a crucial role in determining the length of the intercepted arc and is a key concept in understanding circular geometry. The central angle links the linear measure of the arc (s) and the radius (r) through the arc length formula.

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