Question

If a circular pool has a circular fence around it that has a 20-foot diameter, then what is measure of the central angle that cuts the arc that measures 4 feet

Answer 1

To find the central angle that cuts an arc of 4 feet on a circular fence with a diameter of 20 feet, we can use the formula for the arc length ( \( L \) ) of a circle, which is given by:

\[
L = r \theta
\]

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

First, we need to find the radius of the fence. The radius \( r \) is half of the diameter:

\[
r = \frac{20 \text{ feet}}{2} = 10 \text{ feet}
\]

Given that the arc length \( L \) is 4 feet, we can substitute the values into the arc length formula:

\[
4 = 10 \theta
\]

Now, we can solve for \( \theta \):

\[
\theta = \frac{4}{10} = 0.4 \text{ radians}
\]

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

\[
\theta \text{ in degrees} = 0.4 \times \frac{180}{\pi} \approx 22.918 \text{ degrees}
\]

Thus, the measure of the central angle that cuts the arc measuring 4 feet is approximately:

\[
\theta = 0.4 \text{ radians} \quad \text{or} \quad \theta \approx 22.92 \text{ degrees}
\]