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Formation of a Cone

Date: 03/16/2003 at 21:44:42
From: Star
Subject: Formation of a cone

Why is a cone formed when a sector is removed from a circle and the 
edges of the remaining segment joined?

Does it have to do with the change in proportions?


Date: 03/17/2003 at 10:13:30
From: Doctor Marshall
Subject: Re: Formation of a cone

This is exactly why!

When you take a sector from a circle you aren't changing the radius 
of the shape (it's no longer a circle), but you are reducing the 
partial circumference (i.e., the arc length, or curved perimeter). 

Next the partial circumference is squeezed back into a new smaller 
circle, but the radius is still the same as it was, and the center of 
the (old) circle is still the _same_ _distance_ from every point in 
the new circumference.

So we have a (new) circle with (new) radius r, and some point that is 
equidistant from every point in that circle, namely R units away from 
each point, and R is much bigger than r.

Such a point does not exist in 2-D. We must send our center point out 
into 3-D, as the apex of the cone. The old radius now becomes the 
slant height of the cone. Does this make sense?

Hope this helps. Write back if you have any more questions.

- Doctor Marshall, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Conic Sections/Circles
High School Higher-Dimensional Geometry

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