Question

14 a sector is part of a circle’s area that is defined by a central angle. the ratio of the sector’s area, a, to the circle’s area, \\( \pi r^2 \\), is identical to the ratio of the central angle, \\( \theta \\), to the total measure of the circle, \\( 360^\circ \\).

image of a circle with center p, radius r, and a sector with central angle \\( \theta^\circ \\)

which option represents the formula for the area of a sector?

\\( \boldsymbol{\text{option a}} \\) \\( a = \frac{\theta}{360^\circ} \pi r^2 \\)
\\( \boldsymbol{\text{option b}} \\) \\( a = \theta \pi r^2 \\)
\\( \boldsymbol{\text{option c}} \\) \\( a = 360^\circ \pi r^2 \\)
\\( \boldsymbol{\text{option d}} \\) \\( a = \frac{360^\circ}{\theta} \pi r^2 \\)

Explanation:

Step1: Set up the proportion

The problem states that the ratio of the sector’s area \( A \) to the circle’s area \( \pi r^2 \) is equal to the ratio of the central angle \( \theta \) to \( 360^\circ \). So we can write the proportion:

$$ \frac{A}{\pi r^2} = \frac{\theta}{360^\circ} $$

Step2: Solve for \( A \)

To solve for \( A \), we multiply both sides of the proportion by \( \pi r^2 \). This gives:

$$ A = \frac{\theta}{360^\circ} \cdot \pi r^2 $$

Which matches option A.

Answer:

A. \( A = \frac{\theta}{360^\circ}\pi r^2 \)