Volume of a Hemisphere Using Cavalieri's TheoremDate: 09/09/99 at 09:00:29 From: Theron Pappas Subject: Derive v = (2/3)pi R^3 The Volume of a Hemisphere (a classical application of Cavalieri's theorem): Derive the formula v = (2/3)pi R^3 for the volume of a hemisphere of radius R by comparing its cross sections with the cross sections of a solid right circular cylinder of radius R and Height R from which a solid right circular cone of base radius R and height R has been removed. Date: 09/09/99 at 13:08:02 From: Doctor Peterson Subject: Re: Derive v=(2/3)pi R^3 Hi, Theron. People are often told these days that you need calculus to find the volume of a sphere; but there are several ways to approach it. Here's what you have: --------------------------- ------- ------- +- *****+***** -+ | ------- ***** | ***** ------- | | \ *---------------------------* / | | ** | ** | | * \ | a / * | | ** \ +----------+ ** | | * \ | / * | |* \ |h / *| |* \ | / *| * \ | / * * X---------------------* * solid / \ R * |* / \ *| |* / \ *| | * / \ * | | ** / hollow \ ** | | * / \ * | | ** ** | | / ****---------------------**** \ | | ------- ***** ***** ------- | +- *********** -+ ------- ------- --------------------------- (I've drawn a whole sphere, though your problem deals only with a hemisphere.) If we cut through this figure and look at the cross section, it will look like this: ----------- ----- ----- ---- *********** ---- -- **** **** -- - *** .........../ *** - -- * ... | /... * -- - ** .. | / .. ** - - * . a| / b . * - - * . | / . * - - * . |/ . * - - * . *----------------*----- - * . R. * - - * . . * - - * . . * - - ** .. .. ** - -- * ... ... * -- - *** ........... *** - -- **** **** -- ---- *********** ---- ----- ----- ----------- Here R is the radius of the sphere and of the circumscribing cylinder; a is the radius of a cross-section of the cone at the height where we've cut it; and b is the radius of a cross-section of the sphere. You want to show that the area of the cross-section of the sphere, pi b^2 is the same as the area of the "washer shape' (annulus) that is the cross-section of the cylinder-with-a-cone cut out, pi R^2 - pi a^2 See if you can write equations for a and b in terms of the height h of the cross-section, and then prove what I just said. Cavalieri's theorem will finish the job for you. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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