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Chapter 8: Problem 36

If you walk along the circumference of a circle a distance of 1 diameter, what is the size of the corresponding central angle (in radians)?

Short Answer

Expert verified
The central angle is 2 radians.

Step by step solution

01

Understanding the Problem

We need to determine the central angle, in radians, corresponding to walking a distance equal to one diameter along the circumference of the circle.
02

Recall the Relationship between Radius, Diameter, and Circumference

The diameter of a circle is twice the radius: \( D = 2r \). The circumference of a circle is \( 2\pi r \), or \( \pi D \), since \( D = 2r \).
03

Define the Arc Segment

When walking a distance equal to the diameter along the circumference, we are essentially describing an arc that is equal in length to the diameter \( D \).
04

Formula for Arc Length

The length of the arc \( s \) is related to the central angle \( \theta \) by the formula \( s = r\theta \), where \( r \) is the radius and \( \theta \) is in radians.
05

Substitute Known Values

Given \( s = D = 2r \), substitute into the arc length formula: \( 2r = r\theta \).
06

Solve for Central Angle

Simplify the equation \( 2r = r\theta \) by dividing both sides by \( r \), yielding \( \theta = 2 \). Hence, the central angle \( \theta \) is \( 2 \) radians.

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Key Concepts

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

Radians
Radians are a way to measure angles based on the radius of a circle. When you think of the word "radians," imagine wrapping the radius of a circle around its edge. This measurement is particularly useful because it relates directly to the geometry of the circle.
To understand radians fully:
  • One complete circle is equal to an angle of 360 degrees or \( 2\pi \) radians.
  • A radian is the angle created when the arc length equals the radius of the circle.
  • Therefore, \( \pi \) radians is equal to 180 degrees, making 1 radian approximately 57.3 degrees.
This connection with the radius is why radians come in handy when dealing with circles. For the exercise, we found the central angle was 2 radians. This means the arc length described was twice the radius, indicating a central angle that takes up a bit more than a quarter of the circle. Radians offer a direct method to connect the arc length of a circle and the central angle.
Arc Length
Arc length gives you the distance along a section of a circle's circumference. To find the arc length, you can use the formula \( s = r\theta \), where \( s \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians.
Key points about arc length:
  • The arc length becomes a complete circle's circumference if the angle \( \theta \) is \( 2\pi \) radians.
  • If you know the angle in degrees, convert it to radians first before using the formula.
  • As shown in the exercise, walking a distance equal to the diameter (or \( 2r \)) provides a direct relationship with the central angle in radians, leading to an arc length equal to \( r\cdot2 \).
Arc length helps translate the movement along a circle into measurable terms, simplifying many geometric and practical scenarios.
Circumference
The circumference of a circle is like its perimeter 鈥� the total distance around the edge. It is calculated using the formula \( C = 2\pi r \) or equivalently \( C = \pi D \), where \( D \) is the diameter.
Here are some important aspects of circumference:
  • Circumference helps us understand the entire boundary of a circle.
  • Because a complete circle is \( 2\pi \) radians, knowing the circumference allows us to relate to arc lengths easily.
  • The exercise illustrated moving one diameter along the circumference equates to an angle of 2 radians, emphasizing how part of a circumference can create specific angles.
Understanding circumference provides a basis for calculating many related geometric properties, making sense of how angles and arcs behave in circular contexts.

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