Polygon Inscribed in a Circle

Find the radius, $R$ , of the circumscribing circle

$\pi { R }^{ 2 }\quad =\quad 530.9292\\ \Rightarrow \quad R\quad =\quad 13$

The area of a regular polygon inscribed in a circle is given by

• $n$ is the number of sides
• $r$ is the radius of the circumscribing circle

$A\quad =\quad \frac { n }{ 2 } { r }^{ 2 }\sin { \left( \frac { { 360 }º }{ n } \right) }$

The area of the circle minus the area of the polygon is the area within the circle that is outside the polygon

$\pi { R }^{ 2 }\quad -\quad \frac { n }{ 2 } { R }^{ 2 }\sin { \left( \frac { { 360 }º }{ n } \right) } \quad =\quad 12$

Solving for $n$ , we get

$\frac { n }{ 2 } { R }^{ 2 }\sin { \left( \frac { { 360 }º }{ n } \right) } \quad =\quad \pi { R }^{ 2 }\quad -\quad 12\\ \Rightarrow \quad n\sin { \left( \frac { { 360 }º }{ n } \right) } \quad =\quad \frac { 2\left( \pi { R }^{ 2 }\quad -\quad 12 \right) }{ { R }^{ 2 } } \quad =\quad \frac { 2\left( \pi { \left( 13 \right) }^{ 2 }\quad -\quad 12 \right) }{ { 13 }^{ 2 } } \quad \approx \quad 6.14...$

Among the choices, only $\boxed { 17 }$ works.