# Area of Polygon with integrals

• May 18th 2008, 06:42 PM
CALsunshine
Area of Polygon with integrals
Ok so the question at hand is....

A) Let A[sub-n] be the area of a polygon with 'n' equal sides inscribed in a circle with radius 'r'. By dividing the polygon into 'n' congruent triangles with central angle 2[pi]/n, show that A[sub-n]=1/2[nr<squared>][sin(2[pi]/n).

B) Show that limit as n goes to infinity of A[sub-n]=[pi]r<squared>.

Sorry if my notation is confusing...if you have any questions just let me know....
• May 18th 2008, 07:21 PM
kalagota
Quote:

Originally Posted by CALsunshine
Ok so the question at hand is....

A) Let A[sub-n] be the area of a polygon with 'n' equal sides inscribed in a circle with radius 'r'. By dividing the polygon into 'n' congruent triangles with central angle 2[pi]/n, show that A[sub-n]=1/2[nr<squared>][sin(2[pi]/n).

B) Show that limit as n goes to infinity of A[sub-n]=[pi]r<squared>.

Sorry if my notation is confusing...if you have any questions just let me know....

Your first job is to find the area of each triangle..
Hint.
1. Given a regulan n-gon, what is the measure of each internal angles?
2. Do you think that every side of each triangle is in fact a bisector of the internal angles?
3. Why do you need this? How will you find the altitude/height of each triangle?
4. How will you find the length of the base of each triangle?

This must be sufficient to find the area of each triangle. Now, sum up the area of the triangles and that will be your $A_n$.

Next job is to take the Riemann Sum and let n approach infinity.