In Exercises 7-12, find the radian measure of the central...

In Exercises $7-12,$ find the radian measure of the central angle of a circle of radius $r$ that intercepts an arc of length $s .$
$$\begin{array}{ll}{\text { Radius, } r} & {\text { Are Length,s }} \\ {10 \text { inches }} & {40 \text { inches }}\end{array}$$

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Key Concepts

Radian Measure

Radian measure is a way to quantify angles based on the radius of a circle. One radian is defined as the angle created when the arc length is equal to the radius of the circle, making this measure inherently linked to the geometry of the circle. This unit is particularly useful in mathematics because it simplifies many formulas involving circles and trigonometric functions.

Arc Length Formula

The arc length formula, s = r?, connects the length of an arc (s) with the radius (r) of the circle and the central angle (?) in radians. This formula is fundamental in circle geometry as it allows one to calculate one of these quantities when the other two are known, thereby establishing a direct relationship between linear and angular measurements in circular figures.

Central Angle

The central angle is the angle subtended at the center of a circle by an arc. Its measure in radians directly relates to the length of the intercepted arc through the formula s = r?. Understanding central angles is essential for solving problems related to circular motion and properties of circles in geometry.

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