If the length of an arc of a circle is 44 inches and the radius is 12 inches, then what is the degree measure of the central angle to the nearest tenth of a degree?

1 answer

To find the degree measure of the central angle corresponding to the arc length of a circle, we can use the formula for arc length, which is given by:

\[ L = r \theta \]

where:

  • \( L \) is the arc length,
  • \( r \) is the radius of the circle, and
  • \( \theta \) is the central angle in radians.

We can rearrange this formula to solve for \( \theta \):

\[ \theta = \frac{L}{r} \]

Given:

  • \( L = 44 \) inches,
  • \( r = 12 \) inches,

we can substitute these values into the equation:

\[ \theta = \frac{44}{12} = \frac{11}{3} \text{ radians} \]

Next, we need to convert radians to degrees. The conversion factor from radians to degrees is:

\[ 180^\circ = \pi \text{ radians} \]

Therefore, to convert \( \theta \) to degrees, we use the conversion:

\[ \theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180^\circ}{\pi} \]

Substituting our value of \( \theta \):

\[ \theta_{\text{degrees}} = \frac{11}{3} \times \frac{180^\circ}{\pi} \]

Now we can compute this:

\[ \theta_{\text{degrees}} = \frac{11 \times 180}{3 \pi} = \frac{1980}{3 \pi} \approx \frac{1980}{9.4248} \approx 210.264 \]

Rounding this to the nearest tenth of a degree:

\[ \theta_{\text{degrees}} \approx 210.3^\circ \]

Thus, the degree measure of the central angle is approximately:

\[ \boxed{210.3} \] degrees.