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Day 13
Polygoning the Circle
  
This activity is
taking you back to page 90 "Squaring the Circle" where you found that:
C = kcr and
A = kar2
where
kc
and ka
are specific numbers called
constants of proportionality
You already
developed similar formulas for regular hexagon and the regular octagon.
Comparing the circle
to these circumscribed polygons can give estimates of the value of
kc
and ka .
Notice that the more
sides the polygon has, the better the estimate.
Your
task
In
this activity you will try polygons with different number of sides and
develop formulas for the perimeter and the area of the regular polygon with
that many sides circumscribed about a circle of radius r.
Suggestion:
You can repeat the
process with the following suggested polygons and complete the table below:
| Shape |
Perimeter |
Area |
| Circle |
kcr
= 6r |
ka=
3r2 |
| Square |
8r |
4r2 |
| Regular Hexagon |
6.928r |
3.4641r2 |
| Regular Octagon |
6.63r |
3.31r2 |
| Regular Decagon (10 sides) |
|
|
| Regular Dodecagon (12 sides) |
|
|
| Regular Icosagon (20 sides) |
|
|
| Regular
Pentacontagon (50 sides) |
|
|
| Regular Octacontagon (80 Sides) |
|
|
| Regular Hectogon (100 sides) |
|
|
| Regular Chiliagon (1000 sides) |
|
|
Note: you can find
the name of other polygons
here
Remember: Show your
work and be neat while following these steps.
1. Isolate one of
the triangles.
2, Find and label
the angle < A, the
side S and the
radius should be label r.
3. Isolate one of
the right triangles and label its parts.
New angle =
<A÷2
, new side = x= S
÷2, and the radius = r
(divide the original
angle and the side of the regular polygon by 2)
4. Find the the new
side x
using the tangent of the new angle.
Make sure to use the entire tangent value to
get a good estimation.
5. Use the x values
to find S and the area of the triangle:
S = 2x
A= (b* S)÷2
Finally find the
perimeter and the area of the regular polygon.
Perimeter = # of
sides * S
Area= # of sides * ( Area of one triangle)
Conclusion
After completing the
table make sure to analize by comparing the coefficient of the perimeter
with the coefficient (constant of proportionality) of the area in every
regular polygon.
What do you notice?
Do you recognize the
coefficient (constant of proportionality) in the formula for the area?
What happens to
kc and
ka as the number of sides increases?
Finally, be ready to
present your findings in class.
Mr. Lora
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