Step 1: Problem Definition: A sector's area is given as 1/12th of a full circle's area. The objective is to determine the sector's angle.
Step 2: Sector Area Formula Application: The area \( A_{\text{sector}} \) of a sector with radius \( r \) and angle \( \theta \) (in degrees) is calculated as: \[A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2\]The area of the complete circle is: \[A_{\text{circle}} = \pi r^2\]Given that \( A_{\text{sector}} = \frac{1}{12} \times A_{\text{circle}} \), we can write: \[A_{\text{sector}} = \frac{1}{12} \times \pi r^2\]Substituting the sector area formula: \[\frac{\theta}{360^\circ} \times \pi r^2 = \frac{1}{12} \times \pi r^2\]Canceling \( \pi r^2 \) from both sides yields: \[\frac{\theta}{360^\circ} = \frac{1}{12}\]