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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 103
Chapter 1, Problem 103

Area of a circular sector Prove that the area of a sector of a circle of radius r associated with a central angle θ\(\theta\) (measured in radians) is A=12r2θA=\(\frac\)12r^2\(\theta\).
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Verified step by step guidance
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Start by understanding the relationship between the area of a circle and the area of a sector. The area of a full circle with radius r is given by the formula A = \(\pi\) r^2.
Recognize that a sector is a portion of the circle, defined by a central angle \( \theta \) (in radians). The fraction of the circle's area that the sector occupies is \( \frac{\theta}{2\pi} \), since the full circle corresponds to an angle of \( 2\pi \) radians.
Calculate the area of the sector by multiplying the fraction of the circle's area by the total area of the circle: \( A_{\text{sector}} = \frac{\theta}{2\pi} \times \pi r^2 \).
Simplify the expression: \( A_{\text{sector}} = \frac{\theta}{2\pi} \times \pi r^2 = \frac{\theta r^2}{2} \).
Conclude that the area of the sector is \( A = \frac{1}{2} r^2 \theta \), which is the desired formula.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Central Angle

A central angle is formed by two radii of a circle that meet at the center. It is measured in degrees or radians, with one full rotation around the circle equating to 360 degrees or 2π radians. Understanding central angles is crucial for calculating the area of a sector, as the area is directly proportional to the angle's measure.
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Sector of a Circle

A sector of a circle is a portion of the circle enclosed by two radii and the arc between them. The area of a sector can be calculated using the formula A = (1/2)r²θ, where r is the radius and θ is the central angle in radians. This concept is essential for solving problems related to circular areas and understanding how angles affect the size of the sector.
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Radians

Radians are a unit of angular measure used in mathematics, particularly in calculus and trigonometry. One radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. Radians provide a natural way to relate angles to arc lengths and areas, making them fundamental for deriving formulas like the area of a sector.
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Related Practice
Textbook Question

Design a sine function with the given properties.

It has a period of 1212 with a minimum value of −4-4 at t=0t=0 and a maximum value of 44 at t=6t=6.

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Textbook Question

{Use of Tech} Triple intersection Graph the functions f(x) = x³,g(x)=3^x, and h(x)=x^x and find their common intersection point (exactly).

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Textbook Question

Identify the amplitude and period of the following functions.

q(x)=3.6cos(πx24)q\(\left\)(x\(\right\))=3.6\(\cos\[\left\)(\(\frac{\pi x}{24}\]\right\))

Textbook Question

Beginning with the graphs of y=sinxy=\(\sin\) x or y=cosxy=\(\cos\) x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work.

q(x)=3.6cos(πx24)+2q\(\left\)(x\(\right\))=3.6\(\cos\[\left\)(\(\frac{\pi x}{24}\]\right\))+2

Textbook Question

Identify the amplitude and period of the following functions.

g(θ)=3cos(θ3)g\(\left\)(\(\theta\]\right\))=3\(\cos\[\left\)(\(\frac{\theta}{3}\]\right\))