Editing Spherical cap (section)

==Applications==
===Volumes of union and intersection of two intersecting spheres===

The volume of the [[union (set theory)|union]] of two intersecting spheres
of radii <math>r_1</math> and <math>r_2</math> is
<ref>{{cite journal|first1=Michael L.|last1=Connolly|year=1985|doi=10.1021/ja00291a006|title=Computation of molecular volume|journal= Journal of the American Chemical Society|pages=1118–1124|volume=107|issue=5}}</ref>

:<math> V = V^{(1)}-V^{(2)}\,,</math>

where

:<math>V^{(1)} = \frac{4\pi}{3}r_1^3 +\frac{4\pi}{3}r_2^3</math>

is the sum of the volumes of the two isolated spheres, and

:<math>V^{(2)} = \frac{\pi h_1^2}{3}(3r_1-h_1)+\frac{\pi h_2^2}{3}(3r_2-h_2)</math>

the sum of the volumes of the two spherical caps forming their intersection. If <math>d \le r_1+r_2</math> is the
distance between the two sphere centers, elimination of the variables <math>h_1</math> and <math>h_2</math> leads
to<ref>{{cite journal|doi=10.1016/0097-8485(82)80006-5|year=1982|title=A method to compute the volume of a molecule|journal= Computers & Chemistry|first1=R.|last1=Pavani|first2=G.|last2=Ranghino|volume=6|issue=3|pages=133–135}}</ref><ref>{{cite journal|first1=A.|last1=Bondi|doi=10.1021/j100785a001|year=1964|title=Van der Waals volumes and radii|journal= The Journal of Physical Chemistry|volume=68|issue=3|pages=441–451}}</ref>

:<math>V^{(2)} = \frac{\pi}{12d}(r_1+r_2-d)^2 \left( d^2+2d(r_1+r_2)-3(r_1-r_2)^2 \right)\,.</math>

=== 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.

=== 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>

=== Surface area bounded by parallel disks ===

The curved surface area of the [[spherical segment]] bounded by two parallel disks is the difference of surface areas of their respective spherical caps. For a sphere of radius <math>r</math>, and caps with heights <math>h_1</math> and <math>h_2</math>, the area is

:<math>A=2 \pi r |h_1 - h_2|\,,</math>

or, using geographic coordinates with latitudes <math>\phi_1</math> and <math>\phi_2</math>,<ref>{{cite book|title=Successful Software Development|author=Scott E. Donaldson, Stanley G. Siegel|url=https://books.google.com/books?id=lrix5MNRiu4C&pg=PA354|access-date=29 August 2016|isbn=9780130868268|year=2001}}</ref>

:<math>A=2 \pi r^2 |\sin \phi_1 - \sin \phi_2|\,,</math>

For example, assuming the Earth is a sphere of radius 6371&nbsp;km, the surface area of the arctic (north of the Arctic Circle, at latitude 66.56° as of August 2016<ref>{{cite web|url=http://www.neoprogrammics.com/obliquity_of_the_ecliptic/ |title=Obliquity of the Ecliptic (Eps Mean) |publisher=Neoprogrammics.com |access-date=2014-05-13}}</ref>) is {{math|1= 2''π''{{thinsp}}&sdot;{{thinsp}}6371<sup>2</sup>{{thinsp}}{{abs|sin 90° &minus; sin 66.56°}}}} = {{convert|21.04|e6km2|e6mi2|abbr=unit}}, or {{math|1= 0.5{{thinsp}}&sdot;{{thinsp}}{{abs|sin 90° &minus; sin 66.56°}}}} = 4.125% of the total surface area of the Earth.

This formula can also be used to demonstrate that half the surface area of the Earth lies between latitudes 30° South and 30° North in a spherical zone which encompasses all of the [[tropics]].