In a circle with center \(O\), central angle \(AOB\) has a measure of \(\dfrac{7\pi}{4} \). The area of the sector formed by central angle \(AOB\) is what fraction of the area of the circle?

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Don't overthink this one. An entire circle is \(360 \degree\) or \(2\pi \text{ radians}\). Therefore, the area of the sector to the area of the entire circle is:

$$ \frac{\frac{7\pi}{4}}{2 \pi} $$ $$ = \frac{7 \pi}{4} \cdot \frac{1}{2\pi} $$ $$ = \boxed{\frac{7}{8}} $$

Here's a picture of the situation:

Note that the fraction \(\dfrac{7}{8}\) is the ratio for most corresponding properties of the sector to the entire circle (arc length, sector area, central angle, etc.).