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Chapter 1: Problem 64

The length of an arc intercepted by a central angle of \(\alpha\) radians in a circle of radius \(r\) is _______.

Short Answer

Expert verified
The arc length is \(r \alpha\).

Step by step solution

01

Understand the formula for arc length

To find the length of an arc intercepted by a central angle in a circle, use the formula: \ \[s = r \times \theta\] where \(s\) is the arc length, \(r\) is the radius of the circle, and \(\theta\) is the central angle in radians.
02

Substitute given values into the formula

In this problem, the central angle is given as \(\alpha\) radians and the radius is \(r\). We substitute these into the formula: \ \[s = r \times \alpha\]
03

Simplify the equation

After substitution, the arc length formula becomes: \ \[s = r \alpha\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Central Angle
A central angle is an angle whose vertex is at the center of a circle and whose sides are radii of the circle. It plays a crucial role in determining the arc length and can be measured in degrees or radians.
To visualize, imagine slicing a pizza: the angle at the center, where you start cutting, is the central angle.
Central angles are important in defining portions of the circle, or arcs. For instance, a central angle of 90 degrees will divide the circle into a quarter. Understanding this is key for solving problems involving arc length.
Radians
Radians offer a way to measure angles that relates directly to the circle's circumference. Unlike degrees, which divide a circle into 360 parts, radians define the angle based on the arc length relative to the radius.
Specifically, one radian is the angle created when the length of the arc equals the radius of the circle.
Since the circumference of a circle is \(2\tr \times \text{radius}\text{ ( } r \text{)}\text{, with } \) \frac{360 \text{ degrees equal to } 2 \tr}, a full circle in radians is 2 \tr. This link simplifies many calculations, especially in trigonometry and calculus.
Circle Radius
The radius of a circle is the straight-line distance from the circle's center to any point on its perimeter. This measurement is crucial in determining the length of an arc.
In geometric problems, the radius often serves as a given or an unknown quantity that you must solve for.
When you know the radius and the central angle, you can plug these values into the arc length formula. This makes the radius a key component in finding out important circle properties, including area, circumference, and arc length.
Trigonometry Formula
The arc length formula, \[s = r \times \theta \] \ \r represents the radius and \theta represents the central angle in radians, leverages key trigonometric principles.
This formula essentially states that the length of an arc is a proportion of the circle's circumference: the proportion is the same as that of the central angle to the full angle circle (2 \tr radians).
This makes trigonometric functions and formulas very practical in geometry problems involving circles. Knowing how to manipulate and apply this formula can help you solve a variety of problems concerning the properties of circles.

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