Asked by Matty
Find the perimeter of a regular 360-sided polygon that is inscribed in a circle of radius 5 inches. If someone did not remember the formula for the circumference of a circle, how could that person use a calculator’s trigonometric functions to find the circumference of a circle with a 5-inch radius?
Answers
Answered by
Reiny
Looks like they want you to cut the circle into "many" equal isosceles triangles, finding the base of one of them , then multiplying by the number of such triangles
e.g. in your case , visualize 360 isosceles triangles each with equal sides of 5 and a vertex angle of 1°
The base of one of those can be found using the cosine law:
x^2 = 5^2 + 5^2 - 2(5)(5)cos 1°
= 50 - 50cos 1°
x = √(50 - 50cos1) = .08726...
but you have 360 of these, so the perimeter is
appr 360(.08726...) or 31.41552742
using πD, we get 10π = 31.41592654
which is off by .000399 , not bad!
suppose we increase the number of triangles to 1440, then each angle is .25°
x = √(50-50cos.25) = ....
and perimeter = 1440(.....) = 31.41588633
only off by .00004021 , even better
notice 1440 = 4(360) and 1°/4 = .25
try experimenting with higher multiples
e.g. in your case , visualize 360 isosceles triangles each with equal sides of 5 and a vertex angle of 1°
The base of one of those can be found using the cosine law:
x^2 = 5^2 + 5^2 - 2(5)(5)cos 1°
= 50 - 50cos 1°
x = √(50 - 50cos1) = .08726...
but you have 360 of these, so the perimeter is
appr 360(.08726...) or 31.41552742
using πD, we get 10π = 31.41592654
which is off by .000399 , not bad!
suppose we increase the number of triangles to 1440, then each angle is .25°
x = √(50-50cos.25) = ....
and perimeter = 1440(.....) = 31.41588633
only off by .00004021 , even better
notice 1440 = 4(360) and 1°/4 = .25
try experimenting with higher multiples
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