As the above Figure 1 shows, the experiment at the exhibit “The Circular Surface“ 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 .
is the length of the circumference of the circle that the spring originally formed. Furthermore, the following applies: The area of the right triangle with the 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:
If the two levers are each rotated 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
According to figure 3 above, the following applies to the straight lines and :
Their intersection then results as the solution of the equation
This results in equation (1):
Equations (1) and (2) then give
which after multiplying out and reducing leads to
That now means
Consequently, it can be assumed without restriction 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:
Finally, we give — determined numerically as approximate values — for
() the corresponding angles and the radii (using the above equation).
Table 1: The values and as a function of .
Figure 4 below summarises this in a diagram.