Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.
Get a random Fun Fact!
or
No subject limitations
Search only in selected subjects
    Algebra
    Calculus or Analysis
    Combinatorics
    Geometry
    Number Theory
    Probability
    Topology
    Other subjects
  Select Difficulty  
Enter keywords 

  The Math Fun Facts App!
 
  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su
All rights reserved.

From the Fun Fact files, here is a Fun Fact at the Medium level:

Area of a Circle or Regular Polygon

There's a nice way to see why the formula for the area of a circle of radius R is:


Pi * R2.

It has an interesting relationship with the formula for the circumference of a circle, which is 2 * Pi * R (and that is a consequence of the definition of Pi, which is defined to be the ratio of the circumference of a circle to its diameter.)

So consider a regular polygon, which is an N-sided figure with equal side lengths S and equal angles at each corner. There is an inscribed circle to the polygon that has center C and just barely touches the midpoint of every side. A line from C to the midpoint of a side is called the apothem, and suppose this apothem has length R.

If you cut the polygon along lines from each corner of the polygon to the center C, you will get a bunch of triangles, each with area (1/2)*(base)*(height). Note that each (base) has length S and the (height) is the length R of the apothem, and there are N such triangles. Thus the total area of the polygon is N*(1/2)*S*R, which to say it another way is:


(1/2) (Circumference of the Polygon) * R

Now notice that if you let N, the number of sides of the polygon, get larger and larger, the polygon's area approaches the area of a circle of radius R. On the other hand, the circumference approaches the circumference of a circle, so that as N goes to infinity, the above formula approaches:

(1/2) (2 * Pi * R) * R

which is just Pi*R2, the area of a circle!

Presentation Suggestions:
Draw a circle and cut it into thin pie wedges. Then help students see that each pie wedge is approximated by a very thin triangle, and as we cut the pie into more and more wedges, this approximation gets better and better and in the limit the approximation becomes equality.

The Math Behind the Fact:
The process of approximating the area of the circle by slicing it into thin wedges is analogous to the process of integration (in calculus) to find an area. In the limit this approximation gets better and better. To see the relationship between circumference and area in reverse, where derivatives play a role, see Surface Area of a Sphere. See Rolling Polygons for more connections between polygons and circles.

How to Cite this Page:
Su, Francis E., et al. "Area of a Circle or Regular Polygon." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Keywords:    area of a circle, area of a regular polygon, area of circles
Subjects:    geometry, calculus, analysis
Level:    Medium
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
4.18
 
current
rating
Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!
New: get the MathFeed iPhone App!

Brings you news and views on math:
showcasing its power, beauty, and humanity

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!