To find the approximate area of a regular 13-gon with sides of length 16 yards, we can divide the polygon into 13 congruent triangles, where each triangle has a central angle of 360/13 degrees.
The central angle of each triangle is 360/13 degrees, and the radius of the circumcircle of the polygon is half the length of the diagonal of the 13-gon. To find the diagonal, we can divide the 13-gon into 10 isosceles triangles, where each triangle has base length 16 yards and base angles of 180/13 degrees. We can then use the law of cosines to find the length of the diagonal:
d^2 = 16^2 + 16^2 - 21616*cos(180/13)
d ≈ 176.28 yards
The area of each triangle can be found using the formula:
A = (1/2)bh
where b is the length of the base (16 yards) and h is the height of the triangle. The height of the triangle can be found using the Pythagorean theorem:
h^2 = d^2 - (b/2)^2
h ≈ 174.15 yards
Thus, the area of each triangle is:
A = (1/2)(16)(174.15) ≈ 1394.38 square yards
The total area of the 13-gon is therefore:
A_total = 13A ≈ 18,121.94 square yards
Therefore, the approximate area of the regular 13-gon is 18,121.94 square yards.