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 of the arc bounding a sector with a given area, we can use the formula for the area of a sector given the radius \( r \) and the central angle \( \theta \) (in radians):

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

In this case, we have:
- Area \( A = 10\pi \) square centimeters
- Radius \( r = 5 \) centimeters

Substituting in the known values:

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

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

\[
10\pi = 12.5\theta
\]

Now, solve for \( \theta \):

\[
\theta = \frac{10\pi}{12.5} = \frac{10\pi}{25/2} = \frac{10\pi \cdot 2}{25} = \frac{20\pi}{25} = \frac{4\pi}{5}
\]

Now we want to convert \( \theta \) from radians to degrees. Since \( 180^\circ \) is equivalent to \( \pi \) radians, we can convert it using the following:

\[
\theta \text{ (in degrees)} = \theta \text{ (in radians)} \times \frac{180^\circ}{\pi}
\]

Substituting our value in:

\[
\theta \text{ (in degrees)} = \left(\frac{4\pi}{5}\right) \times \frac{180^\circ}{\pi}
\]

The \( \pi \) cancels out:

\[
\theta \text{ (in degrees)} = \frac{4 \times 180^\circ}{5} = \frac{720^\circ}{5} = 144^\circ
\]

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

\[
\boxed{144^\circ}
\]