What is the area of a regular nonagon with a radius of 14 in.?
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To find the area of a regular nonagon (a polygon with nine sides) with a given radius, we need to divide the nonagon into nine congruent triangles and calculate the area of one of those triangles.
Each triangle in the nonagon has an angle of 40 degrees at the center of the nonagon, since a full circle is 360 degrees and a nonagon has nine sides. In a regular nonagon, all angles at the center are congruent, so each central angle is 360 degrees / 9 = 40 degrees.
The central angle of a triangle is also one of the three angles inside the triangle, and the other two angles are congruent to each other. In a regular nonagon, the sum of the three angles inside a triangle is also 180 degrees. Thus, each of the other two angles is (180 degrees - 40 degrees) / 2 = 70 degrees.
Now, we can use trigonometry to calculate the area of one of these triangles. We can consider one of the congruent triangles to be a right triangle with a right angle and two angles of 70 degrees.
The radius of the nonagon is the hypotenuse of this right triangle. Let's call the base of this right triangle "b" and the height "h". We want to find the area "A" of this right triangle.
Using trigonometry, we can relate the base, height, and hypotenuse of the right triangle using the sine and cosine functions:
sin(70 degrees) = h / 14 in
cos(70 degrees) = b / 14 in
Solving for the height "h" and the base "b":
h = 14 in * sin(70 degrees) ≈ 13.309 in
b = 14 in * cos(70 degrees) ≈ 5.789 in
Now, we can calculate the area of the right triangle using the formula:
A = (1/2) * b * h
A = (1/2) * 5.789 in * 13.309 in ≈ 38.543 in^2
However, we only found the area of one triangle, and there are nine congruent triangles in the nonagon. To find the total area "T" of the nonagon, we multiply the area of one triangle by nine:
T = 9 * A
T = 9 * 38.543 in^2 ≈ 347.887 in^2
Therefore, the area of a regular nonagon with a radius of 14 inches is approximately 347.887 square inches.
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1) Central angle of nonagon: 360 degrees / 9 = 40 degrees
2) Other two angles inside the triangle: (180 degrees - 40 degrees) / 2 = 70 degrees
3) Height of triangle: 14 in * sin(70 degrees) ≈ 13.309 in
4) Base of triangle: 14 in * cos(70 degrees) ≈ 5.789 in
5) Area of triangle: (1/2) * 5.789 in * 13.309 in ≈ 38.543 in^2
6) Total area of nonagon: 9 * 38.543 in^2 ≈ 347.887 in^2