Question

The radius of a circle is 8 meters. What is the angle measure of an arc bounding a sector with area 8​𝜋 square meters?
K=8​𝜋 sq. m
r=8 m
Give the exact answer in simplest form.
°

Answer 1

To find the angle measure of an arc bounding a sector with a given area, we can use the formula for the area of a sector:

\[
A = \frac{1}{2} r^2 \theta
\]

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

We're given:
- \( A = 8\pi \) square meters,
- \( r = 8 \) meters.

Now, substitute the values into the formula:

\[
8\pi = \frac{1}{2} \times (8)^2 \times \theta
\]

Calculating \( (8)^2 \):

\[
(8)^2 = 64
\]

Substituting this back into the equation:

\[
8\pi = \frac{1}{2} \times 64 \times \theta
\]

This simplifies to:

\[
8\pi = 32\theta
\]

Now, solve for \( \theta \):

\[
\theta = \frac{8\pi}{32} = \frac{\pi}{4} \text{ radians}
\]

To convert radians to degrees, we use the conversion factor \( \frac{180^\circ}{\pi} \):

\[
\theta = \frac{\pi}{4} \times \frac{180^\circ}{\pi}
\]

This simplifies to:

\[
\theta = \frac{180^\circ}{4} = 45^\circ
\]

Thus, the angle measure of the arc bounding the sector is:

\[
\boxed{45^\circ}
\]