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Chapter 5: Problem 126

Area of a Sector of a Circle If a slice with central angle \(\alpha\) radians is cut from a pizza of radius \(r,\) then what is the area of the slice?

Short Answer

Expert verified
\( \frac{1}{2} r^2 \alpha \)

Step by step solution

01

Understand the formula for the area of a sector

The area of a sector of a circle is given by the formula \(\frac{1}{2} r^2 \theta\), where \(r\) is the radius of the circle and \( \theta \) is the central angle in radians.
02

Identify the given values

In the problem, the radius \( r \) of the pizza is given as \( r \) and the central angle \( \theta \) is \( \alpha \) radians.
03

Substitute the values into the formula

Substitute the radius \( r \) and the angle \( \alpha \) into the formula for the area of a sector: \[ \frac{1}{2} r^2 \alpha \].
04

Simplify the expression

The area of the slice is simply \[ \frac{1}{2} r^2 \alpha \]. Since no specific values for \(r\) and \( \alpha \) are given, this is the simplified expression for the area of the slice.

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Key Concepts

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

circle
A circle is a perfectly round shape. It is defined as the set of all points in a plane that are at the same distance from a given point called the center. The distance from the center to any point on the circle is called the radius. Understanding the properties of a circle is essential for grasping more complex topics, such as the area of a sector.

Key properties of a circle include:
  • The center, which is the fixed point inside the circle.
  • The radius, which is the distance from the center to the boundary point.
  • The diameter, which is twice the radius and passes through the center, touching two points on the circle's edge.
  • The circumference, which is the total distance around the circle and is given by the formula \( 2 \pi\ r \).
central angle
A central angle is an angle whose vertex is at the center of a circle. The sides, or arms, of the angle extend to the circumference of the circle. This angle is vital when calculating the area of a sector.

Characteristic features of a central angle:
  • It is always formed by two radii of the circle.
  • The measure of the angle determines the size of the sector.
  • For example, in a pizza slice, the central angle is the angle at the tip of the slice, where all the edges converge at the pizza's center.
Remember, the central angle can be expressed in degrees or radians. For calculations involving the area of a sector, it's often simpler to use radians.
radians
Radians are a unit of angular measurement used in many areas of mathematics. One radian is the angle created when the radius is wrapped along the circle's circumference. There are \( 2 \pi \) radians in a full circle, which corresponds to 360 degrees.

Converting between degrees and radians:
  • To convert from degrees to radians: \(\theta (in radians) = \frac{\theta (in degrees) \cdot \pi }{180} \)
  • To convert from radians to degrees: \(\theta (in degrees) = \theta (in radians) \cdot \frac{180}{\pi} \)
Radians simplify the calculation of the area of a sector, which uses the formula \( \frac{1}{2} r^2 \theta \). With \theta in radians, the computation is straightforward.
radius
The radius of a circle is the distance from the center of the circle to any point on its perimeter. The radius is a key component in formulas involving circles, such as the area, circumference, and area of a sector.

Important notes about radius:
  • The radius is always positive and is measured in units such as centimeters, meters, etc.
  • Knowing the radius allows you to calculate the diameter (\diameter = 2 \times radius \).
  • In the area of a sector formula, \( \frac{1}{2} r^2 \theta \, the radius is squared and then multiplied by the central angle in radians.
For example, in a pizza of radius \(r \), if you know \(r \) and the central angle, you can easily find the area of the pizza slice.

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