Question 744·Hard·Math

The circumference is the distance around the circle, so use $C=2\pi r$, where $r$ is the radius from the center to the edge. The given circumference gives $30\pi=2\pi r$. Divide by $2\pi$: $r=15$. The $30$ is not the radius; the circumference formula includes twice the radius.

The circle's area is $A=\pi r^2$. Substitute the radius, $15$: $A=\pi(15)^2$. Type $15^2$ in Desmos. It shows $225$, so $A=225\pi$ square units.

A sector is a slice of the circle. Divide its area by the whole circle's area: $\frac{75\pi}{225\pi}$. Cancel $\pi$: $\frac{75}{225}$. Type $75/225$ in Desmos. It shows $0.333\ldots$; click the fraction button to see $\frac13$. So the sector is one third of the circle.

A full turn is $360^\circ$, so the central angle is one third of that turn. Type $360(1/3)$ in Desmos; it shows $120$. The sector's central angle measures $120^\circ$. Choice C.