Editing Spherical cap (section)

=== Areas of intersecting spheres ===

Consider two intersecting spheres of radii <math>r_1</math> and <math>r_2</math>, with their centers separated by distance <math>d</math>. They intersect if

:<math>|r_1-r_2|\leq d \leq r_1+r_2</math>

From the law of cosines, the polar angle of the spherical cap on the sphere of radius <math>r_1</math> is

:<math>\cos \theta = \frac{r_1^2-r_2^2+d^2}{2r_1d}</math>

Using this, the surface area of the spherical cap on the sphere of radius <math>r_1</math> is

:<math>A_1 = 2\pi r_1^2 \left( 1+\frac{r_2^2-r_1^2-d^2}{2 r_1 d} \right)</math>