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 θ (measured in radians) is A = 1 2 r 2 θ A=\frac12r^2\theta A = 2 1 r 2 θ . <IMAGE>

Welcome back everyone. In the given image R is the radius and alpha is the central angle in radiance for a sector of a circle. Which of the following is the correct equation for the area of the sector of the circle. A says it's R squared alpha divided by two br squared alpha, cr alpha divided by two and D says it's R alpha. Now what do we know about the sector of the circle shaded in red here? Well, the sector of the circle just represents a portion of the circle. So if we can figure out the area for the entire circle, then we can try to find what area will represent that portion of the circle that's covered by the central an alpha. So what do we know about the area of a circle? Well, recall that for the complete circle, its area will be equal to pi multiplied by its radius R squared. And we also know that for the complete circle, theta, the central angle would be equal to two pi radians. No, that means then that when the central angle, OK. So since two pi radians, that's right. I told here two pi radians corresponds to pi R squared. OK? Then one radian if we divide both sides by two pi is going to be equal to pi R squared divided by two pi where Pi cancels pi and leaves us with a half of R squared. That means then that alpha radiance OK is going to be equal to R squared divided by two multiplied by alpha, which is going to be equal to R squared alpha divided by two. Therefore, A is the correct answer. Thanks for watching everyone. I hope this video helped.

0. Functions

Common Functions

Calculus

0. Functions

Common Functions

2:05 minutes

Problem 103

Textbook Question

Area of a circular sector Prove that the area of a sector of a circle of radius r associated with a central angle $\theta$ (measured in radians) is $A=\frac12r^2\theta$ .
<IMAGE>

Verified step by step guidance

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.

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.

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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