Volume of a Spherical CapDate: 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/ |
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