posted by John .
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?
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.