In this assignment, you examine a process that links polygons and circles. You will reach some quantitative conclusions about their respective areas and the relationship between the two.

As you know, a regular polygon has sides of equal length and angles that are the same. This confers a high degree of symmetry to these figures. A circle may be thought of as the logical limit of an n-sided polygon as n goes to infinity; the sides become infinitesimally small, and the interior angle between adjacent sides continues toward 180°.
A circumscribed polygon is a regular polygon that surrounds a circle. The radius of the circle is the length of the apothem of the polygon; both share the same center. This means the circle touches the polygon on the inside in the middle of each side of the polygon. Similarly, an inscribed polygon is surrounded by a circle sharing the same center and goes through each vertice of the polygon. A sample of inscribed and circumscribed pentagons is shown below:

For this exercise, consider a circle of radius 1, and corresponding inscribed and circumscribed polygons with the number of sides n = 3, 4, 5, 6, and 8.
A: For each n = 3, 4, 5, 6 & 8, what are the areas of the inscribed and circumscribed polygons with n sides?
B: As n gets larger, what happens to the ratio of these pairs of areas (use the larger area as the numerator, and the smaller area as the denominator)?
C: Both areas tend toward a limiting value as n gets larger and larger. What number would this be?
D: For each n = 3, 4, 5, 6 & 8, what are the perimeters of the inscribed and circumscribed polygons with n sides?
E: As n gets larger, what happens to the ratio of these pairs of perimeters (use the larger perimeter as the numerator, and the smaller perimeter as the denominator)?
F: Both perimeters tend toward a limiting value as n gets larger and larger. What number would this be?

A: 3=(3/2)sin(2tt/3)=3,3/4 sq units and so on.

3=3*tan(tt/3)=3,3 sq units.

B: go 3.5
C: 3, 6,9, 12.
D: not sure
E: they go up.
F: 6, 12, 18.?

To answer the questions A, B, C, D, E, and F, we need to understand the formulas and concepts related to the areas and perimeters of regular polygons and circles.

A: For each value of n, we need to calculate the areas of both the inscribed and circumscribed polygons.

1. Inscribed polygon: The area of an inscribed regular polygon with n sides can be calculated using the formula:

A_inscribed = (n * r^2 * sin(360°/n))/2

where n is the number of sides, r is the radius of the circle, and sin(360°/n) is the sine of the angle formed by each side.

2. Circumscribed polygon: The area of a circumscribed regular polygon with n sides can be calculated using the formula:

A_circumscribed = (n * r^2 * tan(180°/n))/2

where n is the number of sides, r is the radius of the circle, and tan(180°/n) is the tangent of half the angle formed by each side.

Calculate the areas of the inscribed and circumscribed polygons for each value of n using the above formulas.

B: To analyze the ratio of the areas as n gets larger, compare the values of the areas for larger values of n. Calculate the ratio of the larger area to the smaller area for each pair and observe the trend.

C: As n gets larger and larger, both the inscribed and circumscribed polygons approach the shape of the circle. So their areas tend toward the area of the circle. The area of a circle with radius r is given by:

A_circle = π * r^2

Substitute the value of r from the given radius (1) into the formula to find the limiting value of the areas.

D: The perimeters of both the inscribed and circumscribed polygons can be calculated using the formulas:

1. Inscribed polygon: The perimeter of an inscribed regular polygon with n sides can be calculated using the formula:

P_inscribed = 2 * n * r * sin(180°/n)

2. Circumscribed polygon: The perimeter of a circumscribed regular polygon with n sides can be calculated using the formula:

P_circumscribed = 2 * n * r * tan(180°/n)

Calculate the perimeters of the inscribed and circumscribed polygons for each value of n.

E: To analyze the ratio of the perimeters as n gets larger, compare the values of the perimeters for larger values of n. Calculate the ratio of the larger perimeter to the smaller perimeter for each pair and observe the trend.

F: Similar to the areas, as n gets larger and larger, both the inscribed and circumscribed polygons approach the shape of the circle. So their perimeters tend toward the perimeter of the circle. The perimeter of a circle with radius r is given by:

P_circle = 2 * π * r

Substitute the value of r from the given radius (1) into the formula to find the limiting value of the perimeters.

By following these steps, you can find the answers to the questions A, B, C, D, E, and F.