What is the perimeter of a 30-gon with radius 1
30*2*1*sin(1/2 * 360/30)
look at one of the 30 central triangles.
Each will be isosceles with a vertex angle of 12°
base^2 = 1^2 + 1^2 - 2(1)(1)cos12°
= 2 - 2cos12°
so the perimeter is 30(2 - 2cos12°)
= 60 - 60cos12°
use your calculator if you need a decimal answer.
I messed up, forgot to take the square root of 2-2cos12° for the base of the triangle
perimeter = 30√(2-2cos12) = same answer as oobleck
To find the perimeter of a regular polygon such as a 30-gon, you can use the formula: perimeter = number of sides × length of each side.
In this case, given that the 30-gon has a radius of 1, we need to find the length of each side.
First, let's determine the measure of each interior angle of the regular 30-gon. The formula to calculate the measure of an interior angle in a regular polygon is: measure of each interior angle = (180 × (n - 2)) / n, where n is the number of sides.
So, for a 30-gon, the measure of each interior angle is: (180 × (30 - 2)) / 30 = 168 degrees.
Next, we can use the fact that the measure of each interior angle in a regular polygon is related to the measure of the central angle (formed by connecting the center of the polygon to two adjacent vertices) through the formula: measure of each interior angle = 360 / n, where n is the number of sides.
Thus, the measure of the central angle of the 30-gon is: 360 / 30 = 12 degrees.
Since the radius of the 30-gon is 1, using trigonometry (specifically, the cosine function), we can compute the length of each side. In a right triangle with the radius as the hypotenuse and the adjacent side as the segment connecting the center of the polygon to a vertex, the cosine of the central angle will give us the ratio of the adjacent side to the hypotenuse.
So, the length of each side is: cos(12 degrees) = adjacent side / hypotenuse = x / 1, where x is the length of each side.
Using a calculator, we find that cos(12 degrees) ≈ 0.97814.
Therefore, the length of each side is approximately 0.97814.
Finally, we can find the perimeter of the 30-gon by multiplying the number of sides by the length of each side: perimeter = 30 × 0.97814 ≈ 29.3442.
Hence, the approximate perimeter of the 30-gon with a radius of 1 is 29.3442 units.