Editing Spherical cap (section)

=== Volume of a spherical cap with a curved base ===

The volume of a spherical cap with a curved base can be calculated by considering two spheres with radii <math>r_1</math> and <math>r_2</math>, separated by some distance <math>d</math>, and for which their surfaces intersect at <math>x=h</math>. That is, the curvature of the base comes from sphere 2. The volume is thus the difference between sphere 2's cap (with height <math>(r_2-r_1)-(d-h)</math>) and sphere 1's cap (with height <math>h</math>),

<math>\begin{align}
V & = \frac{\pi h^2}{3}(3r_1-h) - \frac{\pi [(r_2-r_1)-(d-h)]^2}{3}[3r_2-((r_2-r_1)-(d-h))]\,, \\
V & = \frac{\pi h^2}{3}(3r_1-h) - \frac{\pi}{3}(d-h)^3\left(\frac{r_2-r_1}{d-h}-1\right)^2\left[\frac{2r_2+r_1}{d-h}+1\right]\,.
\end{align}
</math>

This formula is valid only for configurations that satisfy <math>0<d<r_2</math> and <math>d-(r_2-r_1)<h\leq r_1</math>. If sphere 2 is very large such that <math>r_2\gg r_1</math>, hence <math>d \gg h</math> and <math>r_2\approx d</math>, which is the case for a spherical cap with a base that has a negligible curvature, the above equation is equal to the volume of a spherical cap with a flat base, as expected.