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January 11th, 2012, 06:32 AM   #1
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Relationship between angle in a cone and circle?

I want to know if there is a relationship between a the angle of the sector of a circle which can be turned into a cone, and the angle inside the cone.
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January 15th, 2012, 08:00 AM   #2
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Re: Relation ship between angle in a cone and circle?





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January 17th, 2012, 12:01 AM   #3
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Re: Relation ship between angle in a cone and circle?

Thanks a lot. The help's much appreciated . I tried finding out experimentally using google sketchup, 3d modelling software. Could'nt find any proper relation between the angles that way though .
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January 17th, 2012, 12:07 AM   #4
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Re: Relation ship between angle in a cone and circle?

Oh, and by the way, are there any sources explaining how the relation is derived
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January 17th, 2012, 12:26 AM   #5
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Re: Relationship between angle in a cone and circle?

Here is my derivation:

Let be the old radius and be the new radius. Then the circumference of the cone is

, where is the angle contained in the sector.

so



The sine of the angle in question is



so the angle is

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January 24th, 2012, 02:06 PM   #6
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Re: Relationship between angle in a cone and circle?

Hello, rozze!

And here is my derivation of the formula . . .


Quote:
I want to know if there is a relationship between the angle of the sector of a circle
which can be turned into a cone, and the vertex angle of the cone.

We have a circle of radius
The diagram shows that a minor sector (less than a semicircle) is removed.[color=beige] .[/color][color=red]**[/color]
The central angle of the remaining major sector is

Code:
            ..*.*.*..
          *:::::::::::*
        *   \:::::::/   *
       *    R\:::::/R    *
              \:::/
      *        \:/        *
      *         *         *
      *                   *
                @
       *                 *
        *               *
          *           *
              * * *

The circumference of the major sector is

The major sector is formed into a cone.
Its slant height is
Let be its base radius.
Let be half its vertex angle.

Code:
              *
             /|\
            / |A\
           /  |  \
        R /   |   \ R
         /    |    \
        /     |     \
       /      |      \
      * - - - + - - - *
          r       r

The circumference of the base is
[color=beige]. . [/color]






~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

[color=red]**[/color]
Of course, the portion removed need not be a minor sector.

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