Cutting a Cylinder out of a SphereDate: 02/25/99 at 06:45:37 From: Minesh Gajjar Subject: Remaining Volume of a cut Sphere I have a geometry problem: A cylindrical hole has been drilled directly through the centre of a sphere. The length of the cylinder is 6 inches. What is the volume remaining in the sphere? Thank you. Minesh Date: 02/25/99 at 15:11:10 From: Doctor Rob Subject: Re: Remaining Volume of a cut Sphere There does not seem to be enough data to solve this problem, yet it does have a solution. In order for this to be true, the solution must be independent of the radius of the cylindrical hole. That means that we can assume that the cylindrical hole has radius zero, and compute the correct answer. Then the diameter of the sphere is 6 inches, and the volume of the sphere will give you the answer. There is a more direct approach using the following diagram, with the cylindrical hole bored horizontally with axis PQ through the center O of the sphere: _..-----.._ .+'-----------`+. ,' |\ 6 | `. ,' | \ | `. / | \R |r \ / r| \ | \ . | \ | . | R-3 | 3 \ 3 | R-3 | P+-------+------+------+-------+Q | | O | | . | | ' \ r| |r / \ | | / `. | | ,' `. | 6 | ,' `+._---------_.+' ''-----'' O is the center of the sphere, R its radius, and r the radius of the cylindrical hole. Then by the Pythagorean Theorem, r^2 = R^2 - 9. See http://mathforum.org/dr.math/faq/formulas/faq.sphere.html and http://mathforum.org/dr.math/faq/formulas/faq.cylinder.html for the formulas used below. The volume of the sphere is 4*Pi*R^3/3. The two missing spherical caps have volume (Pi/6)*(3*r^2+[R-3]^2)*(R- 3), and the cylinder has volume Pi*r^2*6. The remaining volume is then V = 4*Pi*R^3/3 - (Pi/3)*(3*r^2+[R-3]^2)*(R-3) - Pi*r^2*6, = 4*Pi*R^3/3 - (Pi/3)*(3*[R^2-9]+[R-3]^2)*(R-3) - Pi*(R^2-9)*6, = (Pi/3)*(4*R^3 - [R-3]^2*[4*R+6] - 18*[R^2-9]), which simplifies to the correct answer, independent of R or r. For other similar questions in the Dr. Math archives see: What is the Volume of the Sphere? http://mathforum.org/dr.math/problems/gameshow.html Hole in a Sphere http://mathforum.org/dr.math/problems/klein12.30.96.html - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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