Diameter of the Base of a Cone
Date: 08/12/98 at 10:58:25 From: ALAN BROWN Subject: Diameter of the Base of a Cone I am having difficulty remembering and/or finding the formula that would allow me to calculate the diameter of the base of a cone of nine degrees at various lengths. Could you help? The question would be: how many feet wide is the "footprint" of a cone of nine degrees at various depths or distances? Thanks, Alan Brown
Date: 08/12/98 at 14:02:07 From: Doctor Mike Subject: Re: Diameter of the Base of a Cone Hi Alan, This is an interesting question, because you can also ask it without using either the word cone or the word diameter! If you look at the cone from the front (like an ice-cream cone standing upside-down on a table, and your eye is about an inch above the plane of the tabletop) it looks like an isosceles triangle. Isosceles means 2 equal sides. /\ /18\ / \ / \ H feet high / \ / \ / \ /______________\ I am guessing that 18 degrees is the entire angle at the top. Usually in discussions about cones we use the "half-angle" which is the angle between the sloping side and the vertical. If the full angle is 9, the half-angle would of course be 4.5. Let's split the isosceles triangle down the middle to get a right triangle. |\ |9\ | \ | \ H | \ | \ | \ |_______\ R Now we need to introduce some trigonometry into our problems. The tangent of 9 degrees is Opposite over Adjacent, or in this case R/H. Because you know H and want to know R, the most useful version of the formula is: R = H * tan(9) Actually, that would correspond to the Radius of the base of the cone, and what you really want is twice that for the diameter, or: D = 2 * H * tan(9) What's that? You don't know the tangent of 9 degrees? Whip out your calculator or use 0.15838444, which should be close enough. If you need tan(4.5), that is 0.078701707, but go ahead and buy yourself a simple scientific calculator. You can probably get one for under $10. I hope this helps. - Doctor Mike, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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