Question

The radius of a circle is 1 kilometer. What is the area of a sector bounded by a 135° arc?
135°
r=1 km
Give the exact answer in simplest form.

Answer 1

To find the area of a sector of a circle, you can use the formula:

\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\]

where:
- \( A \) is the area of the sector,
- \( \theta \) is the angle in degrees,
- \( r \) is the radius of the circle.

In this case, we have:
- \( \theta = 135^\circ \)
- \( r = 1 \) km

Substituting the values into the formula, we get:

\[
A = \frac{135^\circ}{360^\circ} \times \pi (1 \text{ km})^2
\]

Now, calculate \( (1 \text{ km})^2 \):

\[
(1 \text{ km})^2 = 1 \text{ km}^2
\]

Substituting this back in:

\[
A = \frac{135}{360} \times \pi \, \text{km}^2
\]

Now simplify the fraction \( \frac{135}{360} \):

\[
\frac{135 \div 45}{360 \div 45} = \frac{3}{8}
\]

So the area becomes:

\[
A = \frac{3}{8} \pi \, \text{km}^2
\]

Thus, the area of the sector bounded by a 135° arc in simplest form is:

\[
\boxed{\frac{3\pi}{8} \, \text{km}^2}
\]