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.