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?
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
Well, if someone doesn't remember the formula for the circumference of a circle, they can always use the Pythagorean theorem to calculate the perimeter of the regular 360-sided polygon.
First, they would find the length of one side of the polygon by dividing the circumference of the circle by 360. Since our radius is 5 inches, the circumference can be found using the formula 2 * pi * radius.
Now, if someone wants to use the calculator's trigonometric functions, they can go ahead and do some trigonometry magic! They would take half of the side length, let's call it "a," which is the radius of the inscribed circle. Using the calculator's trig functions, they can calculate the length of the other side, let's call it "b," by finding the value of a times the tangent of half of one of the angles of the polygon (360/2 = 180 degrees).
Once they have the values of "a" and "b," they can use the Pythagorean theorem to calculate the length of the diagonal, which is the hypotenuse. This can be found using the formula: hypotenuse^2 = a^2 + b^2.
Finally, they can multiply the length of one side by 360 to find the perimeter of the polygon.
Or they could just remember the circumference formula: 2 * pi * radius. It's a little less mathy and a lot more straightforward. But what's the fun in that?
To find the perimeter of a regular 360-sided polygon inscribed in a circle of radius 5 inches, we can use trigonometric functions.
1. Identify the central angle of the polygon: Since there are 360 sides, each internal angle would be 360/360 = 1 degree.
2. Determine the length of one side of the polygon in terms of the radius, using trigonometry: Considering the right triangle formed by the radius, half of one side of the polygon, and the center of the circle, we can use the sine function. The formula to find the length of each side is side_length = 2 * radius * sin(internal_angle/2).
3. Calculate the length of one side of the polygon: side_length = 2 * 5 * sin(1/2).
Note: Make sure your calculator is in degree mode.
4. Calculate the entire perimeter of the polygon: Since there are 360 sides, the perimeter will be the product of the side length and the number of sides.
perimeter = side_length * number_of_sides = side_length * 360.
5. Substitute the value you obtained for the side length and calculate the perimeter using a calculator's trigonometric functions.
By following these steps, you can find the perimeter of the regular 360-sided polygon inscribed in a circle with a 5-inch radius without directly using the formula for the circumference of a circle.
To find the perimeter of a regular polygon inscribed in a circle, one can use the formula:
Perimeter = number of sides × length of each side
In this case, we have a regular 360-sided polygon. Each side will have the same length, which we need to find.
To determine the length of each side, we need to calculate the distance between two adjacent vertices (corners) of the polygon. Since the polygon is inscribed in a circle, we can draw lines from the center of the circle to two adjacent vertices. This creates an isosceles triangle with two equal sides and a base equal to the length of one side of the polygon.
Using trigonometric functions, we can find the length of each side of the polygon by calculating the length of one of the equal sides of this triangle. The trigonometric function commonly used in this case is sine.
The formula for the length of each side is:
Length of each side = 2 × radius × sin(180° / number of sides)
In this case, the radius is given as 5 inches, and the number of sides is 360.
Length of each side = 2 × 5 × sin(180° / 360)
Now that we have the length of each side, we can calculate the perimeter of the polygon using the formula mentioned earlier:
Perimeter = number of sides × length of each side
Perimeter = 360 × (2 × 5 × sin(180° / 360))
To use a calculator to find the circumference of a circle with a 5-inch radius, you can follow these steps:
1. Enter the value of 180° on the calculator.
2. Divide the entered value by the number of sides (360 in this case).
3. Take the sine of the obtained value using the calculator's sine function.
4. Multiply the sine value by 2, the radius (5 inches), and the number of sides (360).
5. Finally, you have the perimeter or circumference of the circle.