Archimedes' derivation of the spherical cap area formula Archimedes derived a formula for the area of a spherical cap.
 
so Archimedes says that the curved surface area of a spherical cap is equal to the area of a circle with radius equal to the distance between the vertex at the curved surface and the base of the spherical cap.
$$A = \pi(h^2+a^2)$$
I want to know how Archimedes derived this formula. I have searched on the net and only found solutions using integration. Is there a method to do this without using integration?
 A: Enclose the sphere inside a cylinder of radius $r$ and height $2r$ just touching at a great circle. The projection of the sphere onto the cylinder preserves area.
That is the way Archimedes derived that the area of the sphere is same as lateral surface area of the cylinder which is $= (2 \pi r) (2r)=4\pi r^2$. The projection of the cap on the cylinder has area $(2 \pi r)h$. And since $a^2=r^2-(r-h)^2=2rh-h^2 \Rightarrow 2rh=h^2+a^2$, the area of the cap is $\pi (2rh) = \pi (h^2+a^2).$
Edit: corrected grammar
A: Because a visual solution was requested, I have tried to explain @nsoum 's very correct answer using a diagram.

Consider a horizontal slice of the cylinder of thickness $z$. Let the length of the arc subtended on the sphere be $w$. From the figure, we can see that we can assume the arc to be small enough to be a straight line segment, if the thickness $z$ is small enough.
$$ \frac{x}{r} = \sin(\alpha) = \frac{z}{w} $$
Also, by using these thicknesses we can calculate area of cylindrical slice and spherical slice:
$$ A_{\textrm{cylinder}} = 2\pi r z $$
$$ A_{\textrm{sphere}} = 2\pi x w $$
But we have proved that $xw = rz$, so $ A_{\textrm{cylinder}} = A_{\textrm{sphere}} $.
This can very easily be extended to equating areas of the spherical cap, by slicing the spherical cap into multiple small slices and adding each one's area. We can say that each slice on the sphere will have area equal to the slice on the cylinder, hence area of the spherical cap becomes $2\pi rh = \pi(h^2+a^2)$
A: Made a little visualization of it here. This helps my high school students understand this much better.
https://www.desmos.com/calculator/oxhfbnls9x
A: See "On the Sphere and the Cylinder":


*

*propositions 42 and 43,

*pages 52 and 53 of Heath's 1897 English edition of Archimedes's works,

*pages 244 and 245 of the pdf here. 

