Why isn't the surface area of a sphere with radius r the following:

2*pi * (pi*r)

That comes from the following flow of logic:
Doesn't it makes sense to think of the surface area of the sphere with radius r as the the circumference of the semi-circle with radius r, pi*r (2*pi*r/2 = pi*r), multiplied by 2*pi, effectively "rotating" the semicircle along an orthogonal angular axis 2*pi radians (360 degrees)? In my mind, that would "trace" the surface area of a sphere, wouldn't it? What am I missing / not seeing?

Thanks

the problem is, that to rotate a curve, there has to be a variable radius of rotation. You have the 2pi, but don't specify what the radius of rotation is.

area needs to be in square units, but you only have r once, a linear unit.

The formula you mentioned, 2*pi * (pi*r), is not correct for finding the surface area of a sphere. The correct formula is 4*pi*r^2.

Your logic seems to be based on imagining the surface area of a sphere as the result of rotating a semicircle around an orthogonal axis. While this is a reasonable way to visualize it, the calculation does not match your logic.

To explain why the formula for the surface area of a sphere is 4*pi*r^2, we can consider the following:

1. Imagine slicing the sphere into an infinite number of infinitely thin circles. Each of these circles will have a circumference of 2*pi times its radius.

2. As we move closer to the top of the sphere, the circles become smaller because the distance to the top decreases. At the very top, the circle has a radius of zero.

3. To calculate the surface area, we need to add up the areas of all these infinitesimally small circles. Since the circles closer to the top are smaller, they contribute less to the total surface area.

4. Integrating the areas of the circles from the bottom (radius r) to the top (radius 0), we get the formula 4*pi*r^2, which represents the sum of the areas of all the circles.

So, to find the surface area of a sphere, you multiply the square of the radius by 4*pi.

I hope this explanation clarifies why the formula for the surface area of a sphere is 4*pi*r^2, and why your initial logic did not lead to the correct result. If you have any further questions, feel free to ask!