Problem 97 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the length of the arc on a circle of radius \(r\) intercepted by a central angle \(\theta\). Radius \(r\) 14 inches Central Angle \(\theta\) \(\pi\) radians

Step-by-Step Solution

Answer

The length of the arc is 14\(\pi\) inches.

Step 1: Identify Given Values

The values given in the exercise are the radius \(r\) and the central angle \(\theta\). Here \(r = 14\) inches, and \(\theta\) = \(\pi\) radians.

Step 2: Plug the Values into Arc Length Formula

Using the arc length formula \(S = r \cdot \(\theta\)\), we replace \(r\) with 14 inches, and \(\theta\) with \(\pi\) radians.

Step 3: Compute the Arc Length

Multiply the radius by the central angle: \(S = 14 \cdot \(\pi\) = 14\(\pi\) inches.

Key Concepts

Central Angle

The central angle of a circle is the angle whose apex (vertex) is at the center of the circle and whose sides (legs) extend to the circumference. This angle is pivotal in determining the length of an arc, acting as a proportional segment of the circle based on how large or small the angle is. If a central angle is measured in degrees, a full circle encompasses 360 degrees; if a circle is instead divided using radians, a full circle is equal to \(2\backslash pi\) radians. The measure of the central angle directly dictates the fraction of the circumference that makes up the arc.

For example, an angle of \(90\) degrees, which is a quarter of a full circle, intercepts an arc that is one quarter of the entire circumference. Similarly, an angle of \(\backslash pi\) radians, which is half of a full circle in radian measure, will intercept an arc that is half the circle's circumference.

Radius

The radius of a circle is the straight-line distance from the center of the circle to any point on the circumference. It plays a key role in many geometrical formulas and concepts, including the calculation of the arc length. The radius is constant for a given circle and is often symbolized as \(r\).

In our context, the length of the radius serves as a scalar when calculating arc length. It is proportionally consistent regardless of the size of the central angle—making the radius an integral unit in defining the size of the circle and in effect, dictating the absolute length of the arc that corresponds to a particular central angle.

Radians

Radians are a unit of angle measurement that is based on the radius of a circle. One radian is the angle created when the length of the arc that it intercepts is equal to the length of the radius of the circle. Since the circumference of a circle is \(2\backslash pi r\), there are \(2\backslash pi\) radians in a full circle.

The use of radians provides a natural and direct relationship between the length of an arc and the angle that it spans. By expressing angles in radians for the purposes of arc length calculations, we can effectively employ simple multiplication — the arc length \(S\) is equal to the radius \(r\) times the angle in radians \(\backslash theta\), as in \(S = r \backslash cdot \backslash theta\). This means that when dealing with a central angle of \(\backslash pi\) radians, as in the textbook exercise, we are quite literally saying that the arc length is half the circumference of the circle, since \(\backslash pi\) is half of \(2\backslash pi\), which corresponds to a full turn or complete circle.