You are here:

Geometry/area between concentric polygons

Advertisement


Question
QUESTION: There are 2 concentric polygons (one inside the other).  The outer one is always 100 meters from the boundary of the inner one.  As the inner one increases in area from 0.01 to 100 hectares, at what point does the area in the zone between the two polygons exceed the area of the inner polygon? And is there a formula that would apply to all problems like this?

ANSWER: The area of a polygon with n sides and radius r measured to the center of a side is given by
n*r*tan(180/n).  To find when the area of the outside polygon is twice the area of the inside polgon (making the area of the interior be the same as the area between them), find the value of this formula at r and 100+r.

Doing this gives us 2*n*r*tan(180/n) = n*(r+100)*tan(180/n).
The n*tan() term can be divided out, giving 2r = r+200r+10,000.
This means solving the parabola r - 200r - 10000 = 0.

Use the quadratic equation to find r, the radius of the inner polyugon.


---------- FOLLOW-UP ----------

QUESTION: Just one clarification: My polygon is a circle.  So do I just leave out n (the number of sides) or should I use a differently formulated equation?

Answer
It is not important how many sides the polygon has, it is always true.
Given it has n sides, then it can also have n+1 sides.
Given the last line, n would be allowed to increase indefinitely.
Eventually, n will get so large that it appears to be a circle.

Geometry

All Answers


Answers by Expert:


Ask Experts

Volunteer


Scott A Wilson

Expertise

I can answer whatever questions you ask except how to trisect an angle. The ones I can answer include constructing parallel lines, dividing a line into n sections, bisecting an angle, splitting an angle in half, and almost anything else that is done in geometry.

Experience

I have been assisting people in Geometry since the 80's.

Education/Credentials
I have an MS at Oregon State and a BS at Oregon State, both with honors.

Awards and Honors
I was the outstanding student in high school in the area of geometry and math in general.

Past/Present Clients
Over 8,500 people, mostly in math, with almost 450 in geometry.

©2016 About.com. All rights reserved.