3. Number of sides in a regular polygon with an interior angle of \( 175.2^{\circ} \)

To find the number of sides in a regular polygon with an interior angle of \( 175.2^{\circ} \), we can use the formula for the interior angle of a regular polygon: \[ \text{Interior Angle} = \frac{(n-2) \times 180}{n} \] Setting up the equation: \[ 175.2 = \frac{(n-2) \times 180}{n} \] Solving this will yield \( n = 360/4.8 = 75 \). So, the polygon has 75 sides! Thinking about polygons in a fun way, if your polygon had a party, it would definitely need a spacious dance floor - with 75 sides, that’d be one jam-packed extravaganza! In the world of architecture and engineering, regular polygons with a large number of sides can be highly useful. For instance, creating circular designs or domes often involves these shapes, leading to stunning, symmetrical structures that catch the eye and stand the test of time!