College Geometry

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?

1. 👍 0
2. 👎 0
3. 👁 49
1. 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.?

1. 👍 0
2. 👎 0
posted by Brenda

Similar Questions

1. geometry

I'm having trouble with an assignment that asks me to design a structure so that the sumof the measures of the interior angles of all the individual convex polygons is 4500 degrees. i'm souppoused to have a triangle a hexagon an

asked by Meghan on February 11, 2008
2. Algebra 2

Make a table with a process column to represent each pattern. Write an expression for the number of circles in the Nth figure. The first figure is 5 circles long and 2 circles wide, the second 5 circles long and 3 circles wide.

asked by Tabitha on August 24, 2010

If the corressponding angles of 2 polygons are congruent and the lengths of the corresponding sides of the polygons are proportional the polygons are.. a. Regular b. Congruent c. Symmetrix d. Similar

4. English

what do i write for the intertextual links of a text? Can you amplify your question some? It is not clear on what you are writing, and what you are doing with the intertextual links. i have finished reading a novel, and i have to

asked by Pauline on April 23, 2007
5. MATH

If the corresponding angles of 2 polygons are congruent and the lengths of the corresponding sides of the polygons are proportional, the polygons are... a. Regular b. Congruent. c. Symmetric d. Similar

asked by Dobby on December 7, 2010
6. MATH - circles, circles, circles

Happy Friday Math Gang; I can't seem to wrap my head around this one... If two circles have at most 2 places of intersections, 3 circles have at most 6 places of intersection, and so on... How many places of intersection do 100

asked by MsPi_3.14159265 on July 13, 2018
7. history

i have a history document analysis assignment. For this assignment, i need to discuss the respective worldviews of the British and the Chinese.however, i couldn't understand some of the details after i read the primary

asked by Jc on June 8, 2007
8. 3rd grade math hw help!

hello all, my daughter come home with a math homework "Alphabet Polygons" I have no idea how to use the letters in the alphabet to answer the questions. Please HELP! The questions are as follow: 1) You can make polygons of some of

asked by Kelly on January 31, 2011
9. grammar

In the following sentence select the compound predicate. We could study coins, see jewels, or examine tools. A. study,see B. could study, examine C could study,see, examine D study, see, examine Which of these is the correct

asked by Anonymous on October 9, 2009
10. Geometry

My class is learning about the areas of regular polygons and circles and I'm so confused. How are you supposed to figue out the area a regular triangle inscribed in a circle when all you know is that the radius of the circle is

asked by Sam on April 29, 2011

More Similar Questions