What is the area of a regular decagon with a side length of 7 cm and an apothem of 10.8 cm ? \( \square \) \( 378 \mathrm{~cm}^{2} \) \( 226.8 \mathrm{~cm}^{2} \) \( 453.6 \mathrm{~cm}^{2} \) \( 189 \mathrm{~cm}^{2} \)

To find the area of a regular decagon, you can use the formula: \[ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \] First, calculate the perimeter of the decagon. A regular decagon has 10 sides, so the perimeter \( P = 10 \times 7 \, \text{cm} = 70 \, \text{cm} \). Now, plug this and the apothem into the formula: \[ \text{Area} = \frac{1}{2} \times 70 \times 10.8 = 378 \, \text{cm}^{2} \] So, the area of the decagon is \( 378 \mathrm{~cm}^{2} \). Now, for those curious about shapes, decagons are closely related to other polygons, and their properties can lead to fascinating designs in architecture and art. Did you know that with a little creativity, you can construct a beautiful star using a regular decagon by simply connecting specific non-adjacent vertices? It can be a fun geometric project! If you want to dive deeper, consider the fascinating world of tessellations. Shapes like decagons can tessellate, filling a plane without gaps or overlaps when combined creatively with other shapes. Books by artists like M.C. Escher explore these concepts beautifully, showing how geometry can lead to mesmerizing visual patterns. Happy reading!