Frustum of a ConeDate: 12/09/96 at 00:55:37 From: Bill Blackburn Subject: Cones ? Hi, My question is about cones. I am trying to find the formula for the following: If you take a cone of a given size and cut it from the small end to the large end, then lay it out on a flat surface, what will the inside and outside sizes of this flattened-out cone be? Example: x = diameter of small end of cone x is 3" y = diameter of large end of cone y is 6" z = length of cone z is 2" Any help or suggestions would be greatly appreciated. Thank you. Bill Blackburn Date: 12/11/96 at 21:12:03 From: Doctor Rob Subject: Re: Cones ? Interesting problem! Actually, you seem to be dealing with what is called a "frustum" of a cone. A cone usually refers to a figure formed as follows: Start with a plane and a circle lying in it. The circle is called the base of the cone. Construct the line perpendicular to the plane and passing through the center of the circle. Choose a point, called the vertex, on that line but not in the plane. Now consider all the lines connecting the vertex to points on the starting circle. The distance from the vertex to the center of the base is called the height of the cone. The distance from the vertex to the points on the circle is called the slant-height of the cone. A frustum of the cone is gotten by cutting the cone with a plane parallel to the base and lying between the vertex and the base, and discarding everything on the same side of this cutting plane as the vertex. If I understand your question properly, you get a figure in the plane which looks like this: ______________ /-. d .-\ / `--....--' \ / Pi*x \ s/ \s / \ /________________________\ `-. D .-' `--...________...--' Pi*y The upper arc length is Pi*x, the lower arc length is Pi*y. I have a small problem with your term "length" here. I will assume you mean the height of the frustum, which is the perpendicular distance between the two bases. This is not the same as the slant- height, which would be the length s of the two sides in the above diagram. If you extend the frustum to a full cone, then unroll it in the plane, you would have the same effect as extending the two sides in the above diagram until they meet and point P. Call the angle formed by these two extended lines A. Then the upper and lower arcs are arcs of circles centered at P. If we look at a cross-section of the cone taken through the center line, we would get a diagram like this: P /|\ / | \ / | \ / | \ / | \ / | \ / |h-z \ S / | \ /________|________\ / x/2 | x/2 \ / |z s\ / | \ /____________|____________\ y/2 y/2 If we let h be the height of the entire cone, then using similar triangles, we can get the ratios h/(y/2) = (h-z)/(x/2), which we can solve for h = y*z/(y-x) and h-z = x*z/(y-x). The slant-height of the frustum is then s = Sqrt[z^2 + (y-x)^2/4], which is the length of the two sides in the first diagram. If you want to know the length D of the horizontal line in the first diagram, that will tell you the width of the figure. It can be computed as the slant-height of the full cone times 2*sin(A/2). The slant-height of the full cone from the second diagram is S = Sqrt[h^2 + y^2/4], and the angle A is given by Pi*y/S = Pi*x/(S-s). D = 2*S*sin(Pi*y/(2*S)). Similarly, d = 2*(S-s)*sin(Pi*y/(2*S)). The height of the first diagram, from the line marked d to the arc marked Pi*y, would be given by S - (S-s)*cos(Pi*y/(2*S)). If you don't understand this, or if I have misinterpreted your question, please feel free to write back and ask for further clarification. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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