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 .

Here

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

and

apply.

According to figure 3 above, the following applies to the straight lines and :

and

Their intersection then results as the solution of the equation

i.e.

and thus

This results in equation (1):

and thus

Equations (1) and (2) then give

and thus

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).

(in Radian) | ||||||
---|---|---|---|---|---|---|

Table 1: The values and as a function of .

Figure 4 below summarises 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”.