Question

The circle has center O, and the central angle of the shaded sector measures 135 degrees. The area of the shaded sector is what fraction of the area of the circle?
(The number degrees of an arc in a circle is 360.)

Answer 1

To find the fraction of the area of the shaded sector compared to the area of the circle, we can use the formula for the area of a sector. The area of a sector of a circle is given by:

\[
\text{Area of sector} = \frac{\theta}{360} \times \text{Area of circle}
\]

where \(\theta\) is the central angle in degrees and the area of the circle is calculated as:

\[
\text{Area of circle} = \pi r^2
\]

where \(r\) is the radius of the circle.

For this problem, we know:
- The central angle \(\theta = 135\) degrees.

Thus, we can express the area of the shaded sector as:

\[
\text{Area of shaded sector} = \frac{135}{360} \times \text{Area of circle}
\]

Now, we also need to express the area of the circle. However, since we need the fraction of the area of the sector to the area of the circle, we do not need to calculate the actual area of the circle. We can simply focus on the ratio of the areas:

The fraction of the area of the shaded sector to the area of the whole circle is:

\[
\text{Fraction} = \frac{\text{Area of shaded sector}}{\text{Area of circle}} = \frac{\frac{135}{360} \times \text{Area of circle}}{\text{Area of circle}} = \frac{135}{360}
\]

Now, we simplify \(\frac{135}{360}\):

Both 135 and 360 can be divided by 45:

\[
135 \div 45 = 3, \quad 360 \div 45 = 8
\]

Therefore,

\[
\frac{135}{360} = \frac{3}{8}
\]

So, the shaded sector's area is \(\frac{3}{8}\) of the area of the circle.

Thus, the fraction of the area of the shaded sector compared to the area of the circle is

\[
\boxed{\frac{3}{8}}.
\]