Question

If the combined area of the white and orange regions is 100π, what is the area of ONLY the orange region?

Answer 1

We know the combined area is 100π, and the combined area includes the white section as well as the orange section.

now, if we take the area of the white circle, and "subtract" it from the combined area, we're in effect, making a whole in the larger circle, and what's leftover is just the orange part, because the white would have been subtracted out,

$\bf \textit{area of a white circle}\\\\ A=\pi r^2\quad \begin{cases} r=radius\\ -----\\ r=6 \end{cases}\implies A=\pi 6^2\implies A=36\pi \\\\\\ \stackrel{combined~area}{100\pi }~~-~~\stackrel{white~circle's~area}{36\pi }\implies \stackrel{orange~section}{64\pi }$