Τρίτη, 5 Σεπτεμβρίου 2017

A fun fact

A sphere is cut on the surface into rings as in the diagram. If the vertical height of each of the rings remains the same, say k units. Believe it or not - the surface area of any one of the rings remains constant.
Let $x^2 + y^2 = r^2$ be the circle as in the diagram. We concentrate on the positive-x part of the function. The [0, r] on the x-axis is cut into n equal intervals. (Here are 5.):

The width of the ring is . Then the function is rotated around the x-axis by 4 right angles.
The surface area of the k-th ring, formed by rotating Ik around x-axis, is given by:

Now, $x^2 + y^2 = r^2$, therefore
Substitute (2) in (1), we can get:
Here Sk is independent of k (there is no k in the result). Therefore Sk is a constant.
As there are n rings in positive x direction and n rings in the negative x direction.
The total surface area of all rings is