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

Circular Arcs Find the length $s$ of the circular arc, the radius $r$ of the circle, or the central angle $\theta,$ as indicated. An arc of length $15 \mathrm{ft}$ subtends a central angle $\theta$ in a circle of radius $9 \mathrm{ft.}$ Find the measure of $\theta$ in degrees and radians.

Circular Arcs Find the length $s$ of the circular arc, the radius $r$ of the circle, or the central angle $\theta,$ as indicated.
An arc of length $15 \mathrm{ft}$ subtends a central angle $\theta$ in a circle of radius $9 \mathrm{ft.}$ Find the measure of $\theta$ in degrees and radians.

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Key Concepts

Arc Length Formula

The arc length of a circular segment is determined using the formula s = r?, where s represents the arc length, r is the radius of the circle, and ? is the central angle measured in radians. This formula provides a direct relationship among these three quantities, making it essential for problems involving parts of a circle's circumference.

Central Angle

The central angle is the angle formed at the center of the circle by two radii that intercept an arc. It is a key measurement in circle geometry because it directly influences the length of the intercepted arc and plays an integral role in determining proportions of the circle.

Degree-Radian Conversion

Converting between degrees and radians is an important aspect of trigonometry and geometry since different problems may require different units. The conversion is based on the fact that a full circle is 360 degrees or 2? radians, leading to the relationship 1 radian = 180/? degrees and 1 degree = ?/180 radians.

Radians

A radian is the standard unit of angular measure used in many areas of mathematics. One radian is defined as the angle created when the arc length is equal to the radius of the circle, providing a natural and direct link between linear and angular measurements in circular geometry.

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