Question

The radius of a circle is 5 centimeters. What is the angle measure of an arc bounding a sector with area 10​𝜋 square centimeters?
K=10​𝜋 sq. cm
r=5 cm
Give the exact answer in simplest form.

Answer 1

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

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

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

We know:
- \( A = 10\pi \) square centimeters,
- \( r = 5 \) centimeters.

Substituting this into the formula gives:

\[
10\pi = \frac{1}{2} (5^2) \theta
\]

Calculating \( 5^2 \):

\[
5^2 = 25
\]

Substituting that back in gives:

\[
10\pi = \frac{1}{2} \cdot 25 \theta
\]

Simplifying further:

\[
10\pi = \frac{25}{2} \theta
\]

To isolate \( \theta \), we multiply both sides by \( \frac{2}{25} \):

\[
\theta = \frac{2 \cdot 10\pi}{25}
\]

Simplifying this:

\[
\theta = \frac{20\pi}{25} = \frac{4\pi}{5}
\]

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

\[
\theta = \frac{4\pi}{5} \text{ radians}.
\]