TITLE: how to find surface area of a sphere
QUESTION [5 upvotes]: could any one tell me how to calculate surfaces area of a sphere using elementary mathematical knowledge? I am in Undergraduate second year doing calculus 2. I know its $4\pi r^2$ if the sphere is of radius $r$, I also want to know what is the area of unit square on a sphere.

REPLY [2 votes]: Consider a cylinder radius R intersected by a triangular wedge or prism with length 2R width R thickness t: 
Calculate the area $S$ of the shaded portion by sweeping the arrow from the position shown through $\pi/2$, noting that the height of the shaded portion at angle $\theta$ is $t\cos\theta$ and the width of swept angle $\delta\theta$ is $R \delta\theta$
$$
S = 2 \int_{0}^{\pi/2} t\cos\theta Rd\theta
$$
or
$$
S = 2Rt ^{\dagger}
$$
Now stack multiple trimmed wedges to form a spheroid:

and sum the area of all the shaded portions as $t\rightarrow 0$ noting that $\delta t = R \delta\theta$
$$
A = \int_{0}^{2\pi} 2R R d\theta
$$
or
$$
A = 4\pi R^2
$$
Or more directly $A = \int_{t=0}^{2\pi R} 2Rdt$
$\dagger$ this is the same as the area of the $2R\times t$ face of the wedge which leads to a simple proof that the surface area of a sphere is the same as that of its surrounding cylinder