Surface area of spherical cap by summing up circumferences

Question

I have seen calculations of the surface area of a spherical cap using a surface of revolution approach. Is it possible to instead cut the spherical cap into slices horizontally (imagine the cap is formed by a horizontal slice of the sphere) and then adding them up (integrating them)?

Answer 1

Yes - if you remember that the sides are sloping

Suppose $x$ is the height of a slice from the origin of a sphere radius $r$

Then the circumference of a circle at that height is $2\pi \sqrt{r^2-x^2}$ and a slice thickness $\delta x$ is sloping in with sloping distance $\frac{r}{ \sqrt{r^2-x^2}}\delta x +o(\delta x)$ using the almost similar triangles shown below, so the surface area for that slice is $2\pi r\,\delta x + o(\delta x)$.

It is then a simple integration of $2\pi r$ with respect to $x$ to say that the curved surface area of a spherical cap of height $h$ for a sphere of radius $r$ is $2 \pi r h$

Answer 2

Take the sphere $x^2+y^2+z^2=r^2$ as an example and calculate the area of the cap over the angle $\theta$. The area is integrated over a stack of sliced rings with radius $r\sin t$ and width $rdt $, hence area $ds= 2\pi r^2\sin t dt$ $$A = \int_0^{\theta}ds=2\pi r^2 \int_0^{\theta}\sin t dt= 2\pi r^2(1-\cos\theta) $$