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).
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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”.