Circular Area

Heather Bridges

The students should be familiar with the formula for the Area of Regular Polygon = 1/2 a p

where a = the apothem and p = perimeter. The proof and the discovery of the formula is trivial so it will not be included. The students should also be familiar with the formula for the circumference of a circle C = 2 (pi) r.

Student assignment is to construct inscribed regular polygons with the
number of sides, n, increasing, inside the 6 circles
that is provided by the teacher in there folders. Everyone will have the
same size circles in the classroom so that we can compare information.
Some students may choose to start with n = 4 and some may even go as high
as n = 18 if they can successfully construct the polygons. One suggestion
for doing this is draw in the radius of the circles and then rotate the
radius however to suit the desired polygon. The students should further
be asked to label each circle/polygon with the appropriate area for the
polygons it is only necessary to find the area of the circle once. They
should also label each circle/polygon with n and length of each apothem
and identify the length of the radius of the circles. There is a sample
of the expected student work included.

The next step in the exploration is for the teacher to use a spreadsheet
software to display everyones data in front of the class. Hopefully, by
putting the information together and using all the different values found
from varying n, the students will be able to discover a pattern of what
is happening.

Circular Area

Radius of circle = 0.77 inches

Area of Circle = 1.86 sq. inches

Circumference = 4.72 sq. inches

number of sides Apothem Circumference Area

4 0.54 in 4.25 in 1.19 sq. in.

5 0.62 in 4.42 in 1.41 sq. in.

6 0.67 in 4.51 in 1.54 sq. in.

8 0.71 in 4.60 in 1.68 sq. in

10 0.73 in 4.65 in 1.74 sq. in.

12 0.74 in 4.67 in 1.78 sq in.

At this point, it would be good for the students to get in groups and try
to come up with a patten or an estimation about what will happen when n
becomes larger.

With some probing questions, the students should be able to come to some
certain conclusions about the data.

As n gets larger:

1. The length of the apothem, a, of the polygon is getting closer and
closer to length of the radius of the circle, r.

2. The perimeter of the polygon,P, is getting closer and closer to the
circumference of the circle, C.

3. The area of the polygon, An, is getting closer and closer to the area
of the circle, A.

So the conclusions are:

a -> r

P-> C

An->A

Since we know the for a regular polygon

A = 1/2 a P we can substitute in our conclusions to get

A = 1/2 r C for a circle. We can further use the formula C = 2 (pi) r

A = 1/2 r 2 (pi) r. Which finally gives us

A = pi (r)^2 the formula for circular area

We can now probe the students on their own to investigate further as to
why this works and share their their thoughts to the class.

Return to Great
Theorems Page