Question

The circle has center O, and the central angle of the shaded sector measures 90 degrees. The area of the shaded sector is what fraction of the area of the circle?

Answer 1

To find the fraction of the area of the shaded sector in relation to the area of the entire circle, we can follow these steps:

1. **Area of the Circle:** The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
where \( r \) is the radius of the circle.

2. **Area of the Shaded Sector:** The area \( A_{\text{sector}} \) of a sector is a fraction of the area of the circle, based on the angle of the sector compared to the full circle. Since a full circle is \( 360^\circ \), the formula for the area of a sector with central angle \( \theta \) in degrees is:
\[
A_{\text{sector}} = \frac{\theta}{360} \times A
\]
In this case, the central angle \( \theta \) is \( 90^\circ \).

3. **Calculating the Area of the Shaded Sector:**
Plugging in the values, we get:
\[
A_{\text{sector}} = \frac{90}{360} \times A = \frac{1}{4} \times A
\]

4. **Finding the Fraction:**
Now, we can express the area of the sector as a fraction of the area of the circle:
\[
\frac{A_{\text{sector}}}{A} = \frac{\frac{1}{4} A}{A} = \frac{1}{4}
\]

Thus, the area of the shaded sector is \(\frac{1}{4}\) of the area of the circle.

Therefore, the final answer is:
\[
\boxed{\frac{1}{4}}
\]