Question

A regular polygon inscribed in a circle can be used to derive the formula for the area of a circle. The polygon area can be expressed in terms of the area of a triangle.

Let s be the side length of the polygon,

let r be the hypotenuse of the right triangle,

let h be the height of the triangle, and

let n be the number of sides of the regular polygon.



polygon area = n(1/2sh)



Which statement is true?

A. - As h increases, ns gets closer to 2πr​.

B. - As s increases, ns gets closer to 2πr​.

C. - As r increases, ns gets closer to 2πr

D. - As n increases, ns gets closer to 2πr​.

Answer 1

The true statement is D, which asserts that as the number of sides of the inscribed polygon increases, the perimeter of the polygon approximates the circumference of the circle more closely, ultimately approaching 2πr.

Step-by-step explanation:

The question involves finding the relationship between the perimeter of an inscribed polygon and the circumference of the circle as the number of sides of the polygon increases. We start with a regular polygon inscribed in a circle and examine the perimeter of this polygon as it gains more sides.

For a regular polygon inscribed in a circle, the side length s and the number of sides n are related to the perimeter. The perimeter of this polygon is given by n*s. As the number of sides of the polygon increases, the polygon approximates the circle more closely. This means that as n increases, the product n*s (the perimeter of the polygon) is approaching the circumference of the circle, which is 2πr (where r is the radius).

Therefore, the correct statement is D. - As n increases, ns gets closer to 2πr because with more sides, the sides of the polygon better approximate the curve of the circle. This concept is essential to understanding how calculus and the idea of limits work to find the properties of a circle by using regular polygons.

Answer 2

D. as n increases, ns --> 2πr
h/r cannot increase it is the circle radius
s, side length we want to get smaller and smaller to make a circle