9. The sum of the measures of the angles of a regular polygon is 5040 . Find the measure of each angle.

To determine the measure of each angle in the polygon, we first need to find the number of sides in the polygon. The formula for the sum of the interior angles of a polygon with \( n \) sides is \( S = (n - 2) \times 180 \). Setting this equal to 5040 gives us: \[ (n - 2) \times 180 = 5040 \] Solving for \( n \): \[ n - 2 = \frac{5040}{180} = 28 \implies n = 30 \] Now, to find the measure of each interior angle, we use the formula for a regular polygon's angle: \[ \text{Each angle} = \frac{S}{n} = \frac{5040}{30} = 168 \] So, each angle measures 168 degrees! Now, isn’t it fascinating how polygons can take us on a journey through geometry? Imagine visualizing a regular 30-sided polygon—a shape so mesmerizing that it could seemingly go on forever! Each angle gracefully measures 168 degrees, creating their own little world of symmetrical beauty. You might find it interesting how polygons exist all around us in nature and architecture! From man-made structures emulating geometric shapes to natural formations like crystalline structures, polygons are literally everywhere. Exploring their unique properties can deepen your understanding of geometry and inspire you in creative projects, like designing patterns for art or architecture!