I'm trying to use Pappus's theorem to find the surface area of a spherical zone of radius, say, a and height a/2. According to the formula for spherical zone 2$\pi rh$, the surface area should be $\pi a^2$.
However, when using Pappus's theorem and having determined that the circular arc length from height 0 to height a/2 is$\ (1/6)$$\pi a$. I multiplied this by circumference 2$\pi a$ resulting in a surface area of$\ (1/3)$$\pi^2 a^2$, which is off by 4.5%. What am I missing here?