Volume of a Spherical Cap

Date: 12/06/2000 at 23:41:51
From: Eric Reid
Subject: deriving the volume formula for spherical caps

I know the volume formula for spherical caps but I do not understand 
how it is derived. I believe that the area formula for circles is part 
of the formula, but I am not sure of the rest.  

Thank you for your assistance.

Date: 12/07/2000 at 09:43:51
From: Doctor Peterson
Subject: Re: deriving the volume formula for sperical caps

Hi, Eric.

It would commonly be derived using calculus, but let's see what can be 
done using only other formulas that can be derived geometrically. (We 
have explanations for the volume of a sphere, and for the surface area 
of a sphere, including a spherical cap, in our archives, so I'll use 
those.)

The spherical cap can be seen as the difference between a spherical 
sector and a cone:

                       ********* ------------------------------
                 ******    |    ******            cap
              ***         h|          ***
            **-------------+-------------** ----------
          ** \             |      c      / **
         *     \           |           /     *           sector
        *        \         |         /        *
       *           \    r-h|       /           *  cone
      *              \     |     /  r           *
      *                \   |   /                *
     *                   \ | /                   *
     *                     + --------------------*-------------
     *                                           *
      *                                         *
      *                                         *
       *                                       *
        *                                     *
         *                                   *
          **                               **
            **                           **
              ***                     ***
                 ******         ******
                       *********

The volume of the sector is proportional to the surface area of the 
cap compared to that of the whole sphere (as you can see by picturing 
it as composed of many thin pyramids meeting at the center). Thus it 
is:

                   A_cap                  2 pi r h * 4/3 pi r^3
     V_sector  =  -------- * V_sphere  =  ---------------------
                  A_sphere                      4 pi r^2

                  h * 4/3 pi r^3
               =  --------------  =  2/3 pi r^2 h
                       2 r

(This agrees with what we have in our FAQ - see geometric formulas.)

Now the volume of the cap is:

     V_cap  = V_sector - V_cone

and the volume of the cone is:

     V_cone = 1/3 pi c^2 (r-h)

where c is the radius of the circle where the cone and cap meet, and:

     c^2 = r^2 - (r-h)^2 = 2rh - h^2

Putting this together,

     V_cap = 2/3 pi r^2 h - 1/3 pi (2rh - h^2)(r - h)
           = 2/3 pi r^2 h - 1/3 pi (2r^2h - 3rh^2 +h^3)
           = 1/3 pi h [2r^2 - 2r^2 + 3rh - h^2]
           = 1/3 pi h (3rh - h^2)

The formula in our FAQ uses c (which is "r_1" there) rather than r; we 
can eliminate r by replacing rh with:

     rh = (c^2 + h^2)/2

from the equation above, and get:

     V_cap = 1/3 pi h [3(c^2 + h^2)/2 - h^2]
           = 1/6 pi h [3c^2 + 3h^2 - 2h^2]
           = 1/6 pi h [3c^2 + h^2]

That's the formula.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/