Problem 96 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the radian measure of the central angle of a circle of radius \(r\) that intercepts an arc of length \(s\). Radius \(r\) 80 kilometers Arc Length \(s\) 160 kilometers

Step-by-Step Solution

Answer

The radian measure of the central angle of the circle is 2 radians.

Step 1: Identify the given values

The given values are the radius of the circle (r) which is 80 kilometers and the length of the arc (s) which is 160 kilometers.

Step 2: Apply the formula

Apply the formula to find the radian measure of the angle, which is \(\theta = s/r\). Substitute \(s = 160\) and \(r = 80\) into the formula.

Step 3: Solve the equation

When we plug in the given values we get \(\theta = 160 / 80\) which yields \(\theta = 2\) radians.

Key Concepts

Central Angle

In the context of circle geometry, a **central angle** is an angle whose vertex is at the center of the circle. The sides of this angle are radii of the circle, extending outwards and subtending an arc along the circumference. The central angle is crucial because it determines the size of the arc it intercepts. In this exercise, we're tasked with finding the measure of the central angle in radians.

To find the measure of this angle, we use the relationship between the arc length, the radius, and the central angle given by the formula:

• \( \theta = \frac{s}{r} \)

Here, \( \theta \) is the central angle in radians, \( s \) is the arc length, and \( r \) is the radius of the circle. When the length of the arc \( s \) equals the length of the radius \( r \), the central angle is 1 radian.

Arc Length

The **arc length** is a portion of the circumference of the circle, determined by the central angle. To understand this better, imagine the perimeter of a pizza slice shaped like part of the circle–this curvy edge is essentially what the arc length represents. In the exercise, our arc length is specified as 160 kilometers.

The formula to calculate the arc length \( s \) from the central angle is:

• \( s = r \times \theta \)

Here, \( r \) is the radius, and \( \theta \) is the angle in radians. This formula inherently builds on how many radii fit into an arc subtended by a central angle. By dividing the arc length by the radius, we end up with the radian measure of the angle. A key takeaway is that the total arc length depends on both the size of the angle and the circle’s radius.

Circle Geometry

**Circle geometry** involves the study of the properties and relations of points, lines, and figures in or on circles. It spans across understanding fundamentals such as radii, diameters, circumscribed angles, and most importantly, central angles and arcs. In real-world applications, circle geometry is relevant where circular paths or structures are used.

Key concepts you often encounter include:

• Radius: A constant distance from the center to any point on the circle, pivotal in defining the circle's size.
• Circumference: The total distance around the circle, related to the radius and equal to \( 2\pi r \).
• Diameter: Twice the radius, passing through the center and touching two points on the circle's edge.
• Central Angle and Arc Length: As we've seen, these are closely linked, providing a way to measure portions of the circle.

A deep understanding of circle geometry is essential for solving problems involving any curved paths or rotations. In engineering, design, and even astronomy, these principles help in calculating trajectories, creating gears, and more.