The area of the regular 15-gon with a radius of 10mm is approximately 247.2mm².
Formula for a Regular Polygon Area:
The area (A) of a regular polygon can be calculated using the following formula:
A = 1/2 * n * s^2 * r
where:
n is the number of sides in the polygon (15 for this case)
s is the side length of the polygon
r is the radius of the polygon
Finding the Side Length (s):
We don't directly have the side length (s) of the regular 15-gon. However, we can relate it to the radius (r) using the following formula:
s = 2r * sin(π / n)
In this case:
r = 10mm
n = 15
Substituting these values:
s = 2 * 10mm * sin(π / 15)
s ≈ 8.24mm (round to 2 decimal places)
Calculating the Area (A):
Now that we have the side length (s) and the radius (r), we can calculate the area (A) of the regular 15-gon:
A = 1/2 * 15 * 8.24mm² * 10mm
A ≈ 247.2mm^2 (round to 1 decimal place)
Therefore, the area of the regular 15-gon with a radius of 10mm is approximately 247.2mm².