Find the area of the regular 15-gon with radius 12 mm.
(Do not round until the final answer. Then round to the nearest tenth as needed.)
The formula for finding the area of a regular polygon is:
Area = (1/2) * apothem * perimeter
Where the apothem is the distance from the center of the polygon to the midpoint of one of its sides.
To find the apothem of a regular polygon, we can use the formula:
apothem = radius * cos(180°/n)
Where n is the number of sides of the polygon. In this case, n = 15.
apothem = 12 * cos(180°/15)
apothem = 12 * cos(12°)
apothem ≈ 11.6613 mm
Now we need to find the perimeter of the polygon. Since it is a regular 15-gon, all sides are equal in length. We can calculate the side length using trigonometry:
side length = 2 * radius * sin(180°/n)
side length = 2 * 12 * sin(12°)
side length ≈ 4.1294 mm
Therefore, the perimeter of the regular 15-gon is:
perimeter = 15 * 4.1294
perimeter ≈ 61.9412 mm
Now we can use the formula to find the area:
Area = (1/2) * 11.6613 * 61.9412
Area ≈ 359.4398 mm²
Rounded to the nearest tenth, the area of the regular 15-gon with radius 12 mm is 359.4 mm².