What is the area of a regular dodecagon with a side length of 9 centimeters? Round your answer to the nearest tenth.
+4622 It's better to understand how to find the area of regular polygons: 1/2 * a * p, where a stands for the apothem and p stands for the perimeter of the polygon...
Mark the center and create a triangle joining two of the vertices. From the center of the polygon, drop a perpendicular line(this is the apothem !)
The central angle is 360 / number of sides, so 360/12=30.
Since the perpendicular line cuts the angle of 30 degrees in half, we have two fifteen(15) degree angles..
By the Law of Sines, we have \(\frac{sin 15}{4.5}=\frac{sin 75}{y}\). Solving for \(y\), gives the apothem, which is approximately 16.794 centimeters.
The perimeter of the polygon is 9 * 12 =108 centimeters.
Thus, the answer is \(\frac{1}{2}*16.794*108\approx906.9\) centimeters squared.
