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Question

# If the radius of a circle doubles, will the measure of a sector of that circle double? Will it double if the arc measure of that sector doubles?

Solution

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The area of a sector is given by:

$A=\dfrac{x}{360}\cdot \pi r^2\color{white}\tag{1}$

where $x$ is the measure of the intercepted arc and $r$ is the radius.

Doubling the radius will yield:

$A_r=\dfrac{x}{360}\cdot \pi (2r)^2$

$A_r=\dfrac{x}{360}\cdot \pi (4r^2)$

$A_r=4\cdot\dfrac{x}{360}\cdot \pi r^2$

$A_r=4A$

If the radius is doubled, then the area of the sector is quadrupled, not doubled.

Doubling the arc measure will yield:

$A_a=\dfrac{2x}{360}\cdot \pi r^2$

$A_a=2\cdot \dfrac{x}{360}\cdot \pi r^2$

$A_a=2A$

If the arc measure is doubled, then the area of the sector is doubled.

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