What is the area of a regular 11-gon with a side length of 6 mm and an apothem of 10.2 mm ? Round to the nearest whole number. \( 363 \mathrm{~mm}^{2} \) 337 mm \( { }^{2} \) \( 312 \mathrm{~mm}^{2} \) \( 306 \mathrm{~mm}^{2} \)

To find the area of a regular 11-gon, you can use the formula: \[ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \] First, calculate the perimeter of the 11-gon with a side length of 6 mm: \[ \text{Perimeter} = \text{side length} \times \text{number of sides} = 6 \, \text{mm} \times 11 = 66 \, \text{mm} \] Now, plug the perimeter and the apothem into the area formula: \[ \text{Area} = \frac{1}{2} \times 66 \, \text{mm} \times 10.2 \, \text{mm} = \frac{1}{2} \times 673.2 \, \text{mm}^{2} = 336.6 \, \text{mm}^{2} \] Rounding this to the nearest whole number, you get 337 mm². Can you believe that the ancient Greeks were the pioneers of regular polygons in mathematics? They studied the properties of shapes like the pentagon, hexagon, and even the elusive dodecagon! Their fascination laid the groundwork for geometry as we know it today, making math not just practical but also deeply beautiful. If you’re ever curious about the world of polygons and their many applications, consider diving into architecture or graphic design! Regular polygons are essential in both fields, from the exquisite tessellations of M.C. Escher to the geometric precision in modern buildings. Plus, creating art with these shapes can be a fantastic way to appreciate the math behind them while making something visually stunning.