The measure of an interior angle of a regular polygon is \( 150^{\circ} \). How many sides does the polygon have? The polygon has \( \quad \) sides.

To find the number of sides of a regular polygon when the measure of an interior angle is \( 150^{\circ} \), we can use the formula for the measure of an interior angle of a regular polygon, which is given by: \[ \text{Interior angle} = \frac{(n - 2) \times 180}{n} \] where \( n \) is the number of sides. Setting this equal to \( 150^{\circ} \): \[ \frac{(n - 2) \times 180}{n} = 150 \] Multiplying both sides by \( n \) and simplifying gives: \[ (n - 2) \times 180 = 150n \] \[ 180n - 360 = 150n \] \[ 30n = 360 \] \[ n = 12 \] Thus, the polygon has \( 12 \) sides. The polygon has \( 12 \) sides. Keep in mind that regular polygons can be found in architecture and design, often serving as beautiful and functional elements in everything from museum designs to garden layouts. They provide excellent stability and aesthetics, being frequently used as structural and decorative features. If you're embarking on some polygon-related projects or crafts, double-check your angle calculations! Many stumble upon errors when assuming the number of sides instead of calculating interior angles correctly. A handy tip is to always retouch your formulas before taking a ruler to the paper!