Answer:
Area of Regular Polygon: ( About ) 716.6 m^2; Option B
Step-by-step explanation:
~ In this situation we can apply the formula 1/2 * a * P, provided a ⇒ apothem, and P ⇒ Perimeter of the shape ~
Here we are not directly provided with the apothem, so we have to first plan out our procedure step-by-step:
Let us first divide this dodecagon into 12 triangles, each equilateral provided that this is a regular polygon ( implied ). Now let us draw an altitude to one of these triangles, and provided these are equilateral triangles Coincidence Theorem can be applied to this specific triangle. That would mean this altitude is both an angle bisector and a median, which can help us determine the tan degree of the mini triangle formed by the altitude, and the length of one of the sides of this mini triangle.
1. Knowing that the altitude splits the base of the triangle into the two congruent parts ⇒ one of the congruent parts should be ⇒ 8/2 = 4 meters
2. Now the triangles formed through spliting this figure are all ≅, so one of the angles of the triangle is 360/12 = 30 degrees. The mini triangle formed should thus have a measure of 30/2 as the altitude is an angle bisector ⇒ tan 15.
3. From this you can create a proportion, with a ⇒ apothem:
tan 15/ 1 = 4/a ⇒ tan 15 * a = 4 ⇒ a = 4/tan 15° ⇒ a = ( About ) 14.93
4. The perimeter of this shape would be 8 * 12 ⇒ 96 meters
5. Now let us solve for the area of this regular polygon through substitution into the formula 1/2 * a * P ⇒ 1/2 * 14/93 * 96 = ( About ) 716.64 meters^2
Answer ⇒ Area of Regular Polygon: ( About ) 716.6 m^2
