Derivation of the Formula for the FrustumDate: 08/09/99 at 17:25:29 From: Rizza Subject: Frustum of a Right Circular Cone I am trying to prove that the volume of a frustum of a right circular cone, which is pi*(r^2 + rR = R^2)/3, is equal to the volume of the solid (frustum of a cone) below the plane, where the base of the cone is of radius r and height H is cut by a plane parallel to and h units above the base. My problem is trying to prove that the answer is equal to the equation for the volume of the frustum of a right circular cone given above. Date: 08/09/99 at 22:37:07 From: Doctor Jaffee Subject: Re: Frustum of a Right Circular Cone I think I can help you out. First of all, your formula for the volume of a frustum is not exactly correct. You have an equals sign in the expression which I think is a typographical error. I assume you meant to have an addition sign there. But more importantly, you left the H out of the formula which should have been multiplied by pi. So, the correct formula for the frustum of a right circular cone is pi*H(r^2 + rR + R^2)/3 Now, suppose that you have a cone with a base of radius r and height H and it is cut by a plane parallel to and h units above the base. You want to prove that the volume is pi*h(x^2 + rx + r^2)/3, where x is the radius of the circle where the cone was cut by the parallel plane. Let A be the center of the circle at the base of the cone, B the center of the circle up where the plane sliced the cone, and C the vertex of the cone. Let E be a point on the circle at the base of the cone, and D the point on the segment CE which intersects the circle with center B. You should then see that triangle AEC is similar to triangle BDC, so corresponding sides are proportional. Specifically, (H - h)/H = x/r Cross multiply and get rH - rh = xH Add rh - xH to both sides and get rH - xH = rh which is equivalent to H(r - x) = rh. So, if you divide both sides by r - x, you get H = rh/(r - x). Now, the volume of the frustum is the volume of the original cone minus the volume of the cone that was removed. Use r for the radius and rh(r - x) for the height of the original cone and use x for the radius and rh(r - x) - h for the height of the cone that was removed. If you do the algebra correctly, you will end up with the desired result. Give it a try and write back if my explanation needs clarification or if you have difficulties. - Doctor Jaffee, The Math Forum http://mathforum.org/dr.math/ |
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