Question

If a circle has a 10 foot diameter, then what is the measure of the central angle that cuts the arc that measures 4 feet?(1 point)
Responses

45.8 degrees
45.8 degrees

226.2 degrees
226.2 degrees

72 degrees
72 degrees

22.9 degrees

Answer 1

To find the measure of the central angle that cuts an arc of a circle, we can use the relationship between the arc length, the radius, and the central angle. The formula for the arc length \( L \) is given by:

\[
L = r \cdot \theta
\]

where \( r \) is the radius and \( \theta \) is the central angle in radians.

First, we need to find the radius of the circle. Given that the diameter is 10 feet:

\[
r = \frac{\text{diameter}}{2} = \frac{10 \text{ feet}}{2} = 5 \text{ feet}
\]

We know the length of the arc \( L \) is 4 feet. Now we can substitute the values into the formula and solve for \( \theta \):

\[
4 = 5 \cdot \theta
\]

To find \( \theta \):

\[
\theta = \frac{4}{5} = 0.8 \text{ radians}
\]

Now, we need to convert the angle from radians to degrees. We can do this using the conversion factor \( \frac{180 \text{ degrees}}{\pi \text{ radians}} \):

\[
\theta_{\text{degrees}} = 0.8 \cdot \frac{180}{\pi} \approx 0.8 \cdot 57.2958 \approx 45.8 \text{ degrees}
\]

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

**45.8 degrees**.