Area of a sector. Game on.

The area of the whole circle is $\pi r^{2} = \pi \cdot 6^{2} = 36\pi\textsf{.}$ A $90^{\circ}$ sector is $\dfrac{90}{360} = \dfrac{1}{4}$ of the circle, so the area is $\dfrac{1}{4} \cdot 36\pi = 9\pi\text{ m}^{2}\textsf{.}$

The area of the whole circle is $\pi r^{2} = \pi \cdot 4^{2} = 16\pi\textsf{.}$ A $60^{\circ}$ sector is $\dfrac{60}{360} = \dfrac{1}{6}$ of the circle, so the area is $\dfrac{1}{6} \cdot 16\pi = \dfrac{8\pi}{3}\text{ in}^{2}\textsf{.}$

The area of the whole circle is $\pi r^{2} = \pi \cdot 6^{2} = 36\pi\textsf{.}$ A $120^{\circ}$ sector is $\dfrac{120}{360} = \dfrac{1}{3}$ of the circle, so the area is $\dfrac{1}{3} \cdot 36\pi = 12\pi\text{ ft}^{2}\textsf{.}$

The area of the whole circle is $\pi r^{2} = \pi \cdot 10^{2} = 100\pi\textsf{.}$ A $90^{\circ}$ sector is $\dfrac{90}{360} = \dfrac{1}{4}$ of the circle, so the area is $\dfrac{1}{4} \cdot 100\pi = 25\pi\text{ mm}^{2}\textsf{.}$

The area of the whole circle is $\pi r^{2} = \pi \cdot 8^{2} = 64\pi\textsf{.}$ A $90^{\circ}$ sector is $\dfrac{90}{360} = \dfrac{1}{4}$ of the circle, so the area is $\dfrac{1}{4} \cdot 64\pi = 16\pi\text{ yd}^{2}\textsf{.}$

The area of the whole circle is $\pi r^{2} = \pi \cdot 4^{2} = 16\pi\textsf{.}$ A $90^{\circ}$ sector is $\dfrac{90}{360} = \dfrac{1}{4}$ of the circle, so the area is $\dfrac{1}{4} \cdot 16\pi = 4\pi\text{ km}^{2}\textsf{.}$

The area of the whole circle is $\pi r^{2} = \pi \cdot 3^{2} = 9\pi\textsf{.}$ A $120^{\circ}$ sector is $\dfrac{120}{360} = \dfrac{1}{3}$ of the circle, so the area is $\dfrac{1}{3} \cdot 9\pi = 3\pi\text{ dm}^{2}\textsf{.}$

The area of the whole circle is $\pi r^{2} = \pi \cdot 12^{2} = 144\pi\textsf{.}$ A $60^{\circ}$ sector is $\dfrac{60}{360} = \dfrac{1}{6}$ of the circle, so the area is $\dfrac{1}{6} \cdot 144\pi = 24\pi\text{ m}^{2}\textsf{.}$