Finding Central Angle from Arc Length

We are asked to find the degree measure of the angle subtended at the center of a circle given the circle's radius and the length of the arc.

Given:

• Radius of the circle, $r = 100$ cm
• Arc length, $L = 22$ cm

Calculating Angle in Radians

The formula relating arc length ($L$), radius ($r$), and the central angle in radians ($\theta_{rad}$) is:

$ L = r \times \theta_{rad} $

Rearranging the formula to solve for $\theta_{rad}$:

$ \theta_{rad} = \frac{L}{r} $

Substituting the given values:

$ \theta_{rad} = \frac{22 \text{ cm}}{100 \text{ cm}} = 0.22 \text{ radians} $

Converting Radians to Degrees

To convert an angle from radians to degrees, we use the conversion factor $\frac{180^{\circ}}{\pi}$.

$ \theta_{deg} = \theta_{rad} \times \frac{180^{\circ}}{\pi} $

We can approximate $\pi$ as $\frac{22}{7}$.

$ \theta_{deg} = 0.22 \times \frac{180^{\circ}}{22/7} $

$ \theta_{deg} = \frac{22}{100} \times \frac{180 \times 7}{22} $

Cancel out the 22s:

$ \theta_{deg} = \frac{1}{100} \times 180 \times 7 $

$ \theta_{deg} = \frac{180 \times 7}{100} = \frac{18 \times 7}{10} = \frac{126}{10} $

$ \theta_{deg} = 12.6^{\circ} $

Expressing Angle in Degrees and Minutes

The angle is $12.6$ degrees. The decimal part ($0.6$) needs to be converted into minutes.

There are 60 minutes in a degree ($1^{\circ} = 60'$).

Minutes $= 0.6 \times 60'$

Minutes $= 36'$

Therefore, the angle in degrees and minutes is $12^{\circ} 36'$.