Construction of Cone SurfaceDate: 10/12/2002 at 11:00:19 From: Albert Subject: Construction of cone surface What is the relation between the angle of a cone at the vertex and the flat angle (theta) of the developed surface on a plane? Example: I want to build a cone with a 30-degree aperture angle at the vertex, height h, and base radius r. What is the angle at the vertex of the circle sector when I develop that surface on the floor prior to cutting the material? Is there a general formula or function theta = f(alpha)? Alpha = angle at the vertex. Thank you. Date: 10/12/2002 at 23:22:10 From: Doctor Peterson Subject: Re: Construction of cone surface Hi, Albert. This is a common question; I searched our site for "cone surface sector" and found this, among others, which gives a picture: Lateral Surface of a Cone http://mathforum.org/library/drmath/view/55082.html Let's copy that picture and add a side view of your cone: ********* ****** ****** *** / ** / ** / * / * /s * / * / * / * / * +---------------------* * theta * * * * * * * * * * * ** ** ** ** *** *** 2 pi r ****** ****** ********* + /|\ / | \ /alpha\ / | \ / | \s / |h \ / | \ / | \ / | \ +---------+---------+ r r The angle theta can be found by taking the ratio of the arc length, 2 pi r, to the circumference of the whole circle, 2 pi s, and multiplying 360 degrees by that ratio. This is the angle r/s times 360 degrees. Since your angle alpha satisfies sin(alpha/2) = r/s you just have to multiply 360 degrees by this sine. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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