Question

A student claims that when you double the radius of a sector while keeping the measure of the central angle constant, then you double the area of the sector. Do you agree or disagree?

Answer 1

In order to find if the student is correct or not, let's analyze the formula for the area of a circular sector:

$A=\frac{r^2\theta}{2}$

Where r is the radius and theta is the central angle.

If the radius is doubled, since it is squared, the area will be multiplied by 4 instead of being doubled as well:

$\begin{gathered} r^{\prime}=2r \\ A^{\prime}=\frac{r^{\prime}^2\theta}{2}=\frac{(2r)^2\theta}{2}=\frac{4r^2\theta}{2}=4A \end{gathered}$

Therefore we disagree with the student.