### Classroom Activity Sheet on Polygonal Drums

Adapted from p. 299 of
"Mathematics: A Human Endeavor", by Harold R. Jacobs, 2nd edition
Most drums make sounds with no definite pitch. The kettledrum,
however, can be tuned to play notes of different pitches by
adjusting the tension
on the skin that is its head.

Imagine that we have a set of four special kettledrums whose heads have the
shape of
regular polygons and an ordinary drum whose head is circular in shape.
If all the heads of these drums have the same area and the same tension,
then their pitches (or frequencies) will vary according to their shape.

Assume that the **pitches of the drums** are as follows, where
pitch is in units of vibrations per second.

Equilateral triangle, 146

Square, 136

Regular hexagon, 132

Regular octagon, 131

Circle, 130
1) What happens to the pitch of a regular polygon drum as the number of
sides of its head increases?

Do you think that regular polygon drums could be built having the
same area and tension as do the drums listed in the table but having
approximately the following pitches? Explain why or why not?

2) 140 vibrations per second

3) 134 vibrations per second

4) 125 vibrations per second

Assume that all of these drums have an **area of 1 square foot**.
1) What are the lengths of the sides of each figure?** Show your work!**

Square

Equilateral Triangle

Hexagon

Octagon

Answers:
equilateral triangle 2/(sqrt(sqrt(3)) ~ 1.519671371, square 1,
hexagon sqrt(2)/sqrt(sqrt(27)) ~ .6204032393, octagon ~.5946035574
2) Explain why it makes sense that as the number of sides increases,
then the side length should decrease?

3) What should happen to the side length as the number of sides
approaches infinity? What would the side length approach? What
would the figure look like?

Assume instead
that all of these drums have a **perimeter of 1 foot**.
1) What is the area of each of the figures?

Square

Equilateral Triangle

Hexagon

Octagon

Answers:
equilateral triangle sqrt(3)/36 ~ .04811252245, square .0625,
hexagon sqrt(3)/24 ~ .07216878367,
octagon 1/32*sqrt(2+sqrt(2))/(sqrt(2-sqrt(2)))
.07544417378,
2) What happens to the area as the side length increases?

3) What is the area of a circle of circumference 1? Show your
work.

Answer: 1/(4Pi) ~ .07957747153

4) Explain why a circle is the shape that maximizes the area
of an enclosed regular region when using a given amount of fencing.