# The compressed circle

As the preceding Figure 1 shows, the experiment at the exhibit *“The Compressed Circle”* consists of bending up a *circular* spring by means of two (initially vertical and parallel) rotatable levers in such a way that its shape in the final position (see Figure 2 below) represents a distance from point to point .

Here

is the *length of the circumference of the circle* that the spring originally formed. Moreover, the *area* of the right triangle with vertices , and (see Figure 1) is equal to half the area of the circle, i.e.

Thus, the area of a circle with radius can be represented by the sum of the areas of two *congruent* triangles.

#### And now … the mathematics of it:

If the two levers are turned by the angle against the vertical axis, the *contact point* between the right lever arm and the bent-up circle results on the right side. The latter now represents itself as a *circular arc* with the *radius* and the *opening angle* (in radians!). Thus

und

According to Figure 3 above, for the straight lines and ,

and

Their intersection is then obtained as the solution of the equation

i.e.

and thus

According to equation (1), this results in

und somit

Equations (1) and (2) then yield

and thus

which, after multiplication out and truncation, leads to

leads. This is now called

(1)

Consequently, without restriction, it can be assumed that , so that for a given opening angle (in radians) of the right lever (see Figure 3), the opening angle of the corresponding circular arc (with radius ) is obtained as the solution of the following equation:

(2)

Finally, we give — determined numerically as approximations — for () the corresponding angles and the radii (using the above equation).

(in Radian) | ||||||

Figure 4 below summarizes this in a diagram.

*Note:* This exhibit is closely related to the exhibits “What is Pi?”, “What is the area of a circle?”, and “Twelve Corners”.