Final answer:
The area of the top of the dodecagon cookie is 403.308 cm².
Step-by-step explanation:
To find the area of the top of the dodecagon cookie, we first need to determine the length of one of its sides. Since the dodecagon has 12 sides, we divide the perimeter by 12 to find the length of one side. The perimeter of the dodecagon is 12 multiplied by the length of one side, so in this case, the perimeter is equal to 6 cm multiplied by 12, which equals 72 cm. Therefore, the length of one side of the dodecagon cookie is 72 cm divided by 12, which is equal to 6 cm.
Now we can calculate the area of the top of the cookie. The dodecagon can be split into 12 congruent isosceles triangles, each with a base equal to one side of the dodecagon and a height equal to the apothem (the line from the center of the dodecagon to the midpoint of a side). The apothem can be determined using the formula: apothem = side length / (2 x tan(180° / number of sides)). In this case, apothem = 6 cm / (2 x tan(180° / 12)), which simplifies to 6 cm / (2 x tan(15°)). Using a calculator, we find that tan(15°) is approximately 0.2679. Therefore, the apothem is 6 cm / (2 x 0.2679), which is equal to 11.203 cm.
The area of each isosceles triangle can be calculated using the formula: area = (base x height) / 2. Plugging in the values, we get area = (6 cm x 11.203 cm) / 2, which simplifies to 33.609 cm². Since there are 12 isosceles triangles forming the dodecagon, the total area of the top of the cookie is equal to the area of one isosceles triangle multiplied by 12, which is 33.609 cm² x 12, resulting in an area of 403.308 cm².