On Computing The Surface Area of a Sphere

- L. Van Warren

If you ask the average calculus student to write a formula for the area of sphere, you will likely get:

where r is the distance from the sphere center to any point on the surface and pi is a trancendental number of infinite decimal length (around 3.1416).

This equation can be factored into the equivalent equation:

But we recognize that this is simply:

 

where d is the diameter of the sphere.

Recognizing that pi times the diameter is just the circumference c, we have:

Thus the area of a sphere is the diameter times the circumference.


When we compute the area of a flat, square piece of carpet, we multiply the width by length.

In the case of the sphere, the width is the diameter, and the length is the circumference.

Here it is in a perspective view:

Here it is from the front.

Mathematically the green cylinder, has exactly the same surface area as the sphere.

(There are no end caps on the cylinder, it is just a surface!)

If you push the green cylinder material over onto the sphere, it only goes 64% of the way around, like so:

The cylinder material is not extensible. It cannot be "stretched" to fit, only cut into smaller bits to cover the exposed areas.

So the following picture is missing something:

The question is, "How do the ends get covered?"

The answer is that as you fold the ends over, there is extra material left as darts, folds and buckled regions.

Those extra folds, when cut and placed contain exactly the area needed to cover the ends missed.

Notice the points of buckling near the middle of the sphere. There is extra material there also.
I cannot easily do the buckled rendering on my computer.