Part 1. Making an involute.

Think of a polygon cut out from a piece of wooden board. Treat it as a spool, and put a thread around it at least once, but possibly several times.
Now tie a pencil to one end of the thread, and fix the polygon at the center of a large sheet of paper. Start unwinding the thread, drawing a spiral around the polygon. This spiral is an involute of the polygon. (see pictures).

    

An involute of a polygon is made from arcs of circles. Therefore it can be drawn with a compass and ruler. We illustrate it below by giving instructions on how to draw the involute of an irregular pentagon. (see picture).

• Draw a (small) pentagon ABCDE, and extend each edge into a ray clockwise.
• Put the point of your compass at A and make a counterclockwise arc E-1.
• Put the point of your compass at B and make a counterclockwise arc 1-2.
• Put the point of your compass at C and make a counterclockwise arc 2-3.
• Put the point of your compass at D and make a counterclockwise arc 3-4.
• Put the point of your compass at E and make a counterclockwise arc 4-5.
• You may continue this process until your paper is no longer big enough to hold the arcs! We have drawn two more:
• Put the point of your compass at A and make a counterclockwise arc 5-6.
• Put the point of your compass at B and make a counterclockwise arc 6-7.

Practice this technique for a while, drawing involutes of different polygons (not just regular ones), until you are familiar with their sizes and shapes.

To see a animation of an involute of a circle click here

Task

Draw an involute of a polygon on a large piece of poster board that is colored only on one side. Cut out the polygon, and cut out all the sections of circles. Can you rearrange them again? In how many ways? What other shapes can you make from them? From the picture above you would get:

A pentagon, ABCDE.
Five complete sections of circles, AE1, B12, C23, D34, E45.
Two incomplete sections of circles, 1E56 and 2167.

Here are some involutes made by students:

Part 2. Finding the length of the involute of a polygon We are going to learn to compute the length of an involute of a polygon. First we will do it mathematically, and then we will do it with string.

• The involute consists of a series of arcs of circles of varying radii.
• The radii are found by measuring the lengths of the edges of the polygon.
• The lengths of the arcs are found by measuring the exterior angles of our polygon.

We will show how to find the length by working through two examples.

1. Finding the length of an arc of a circle (see picture ).

We have drawn a piece of a circle, and we have measured its radius r = 4.5 cm, , and the angle b = 44 degrees. We want to know its arc length P. If P were the circumference of a whole circle, its length would be equal to 2*pi*r, the formula for the circumference of a circle. But P goes only 44/360 of the way around the circle, so its length is

P = (44/360)*2*pi*4.5


Using a TI-108, we get
[44][/][360][*][2][*][3.14][*][4.5][=] display: 3.4529993
P is about 3.5 cm long.

2. Finding the length of an involute mathematically (see picture).

We use the technique in (1) above to find the lengths of the arcs L1, L2, L3, L4, L5, L6, and L7 of the involute. We measure the five exterior angles and the five sides of the polygon.

(Note that the sum of the exterior angles of the polygon is 360 degrees.)

As in (1) above, the length of each arc is ((measure of angle)/360)*2*pi*(length of radius) = pi *(2/360)*(measure of angle)*(length of radius).

We can simplify this formula by substituting 1/180 for 2/360. When we do this, we get:
length of an arc = (pi/180)*(measure of angle)*(radius)

We need to compute the lengths of each of the arcs, and add up all the lengths to get the total length of the involute.

The radii of the sectors grow:

(Do you see why arc L6 has a radius that consists of 2*s1 and why arc L7 has a radius that consists of 2*s1 + 2*s2 ?)

We can factor out (pi/180) from each of the lengths in the rightmost column, when we add them up to get the total length.

The length of the involute is
(pi/180)(60*1.9 + 41*3.2 + 52*4.4 + 106 *6.4 + 101*9.7 + 60*11.6 + 41*12.9)

On the TI-108,

[60][*][1.9][M+]
[41][*][3.2][M+]
[52][*][4.4][M+]
[106][*][6.4][M+]
[101][*][9.7][M+]
[60][*][11.6][M+]
[41][*][12.9][M+]
[MRC][*][3.14][/][180][=] display: 58.561

The involute is about 58.5 cm long.

3. Finding the length using string.

Tie a knot in the end of a string, and lay the string over the arcs of the involute, putting the knot at one end of the involute. When you get to the other end of the involute, mark the string with a colored marker. Measure the length of the string from the knot to the mark. It should be about 58.5 cm long!