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 January 11th, 2012, 06:32 AM #1 Newbie   Joined: Jan 2012 Posts: 3 Thanks: 0 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.
 January 15th, 2012, 08:00 AM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,950 Thanks: 1141 Math Focus: Elementary mathematics and beyond Re: Relation ship between angle in a cone and circle? $\lambda\,=\,\sin^{\small{-1}}$$1\,-\,\frac{\theta}{2\pi}$$$ $\text{where }\lambda\text{ is the angle between the height from base to apex and the slanted side and }\theta\text{ is the angle of the sector.}$ $\text{Angles are in radians.}$
 January 17th, 2012, 12:01 AM #3 Newbie   Joined: Jan 2012 Posts: 3 Thanks: 0 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 .
 January 17th, 2012, 12:07 AM #4 Newbie   Joined: Jan 2012 Posts: 3 Thanks: 0 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
 January 17th, 2012, 12:26 AM #5 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,950 Thanks: 1141 Math Focus: Elementary mathematics and beyond Re: Relationship between angle in a cone and circle? Here is my derivation: Let $r$ be the old radius and $R$ be the new radius. Then the circumference of the cone is $2\pi r\,-\,r\theta$, where $\theta$ is the angle contained in the sector. so $r(2\pi\,-\,\theta)\,=\,2\pi R\,\Rightarrow\,R\,=\,$$1\,-\,\frac{\theta}{2\pi}$$r$ The sine of the angle in question is $\frac{R}{r}\,=\,1\,-\,\frac{\theta}{2\pi}$ so the angle is $\sin^{\small{-1}}$$1\,-\,\frac{\theta}{2\pi}$$$
January 24th, 2012, 02:06 PM   #6
Math Team

Joined: Dec 2006
From: Lexington, MA

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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 $R.$
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 $\theta.$

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

The circumference of the major sector is $R\theta.$

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

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

The circumference of the base is $R\theta.$
[color=beige]. . [/color]$2\pi r \,=\,R\theta \;\;\;\Rightarrow\;\;\;r \,=\,\frac{R\theta}{2\pi}$

$\sin A \;=\;\frac{r}{R} \;=\;\frac{\frac{R\theta}{2\pi}}{R} \;=\;\frac{\theta}{2\pi}$

$\text{Therefore: }\:A \;=\;\sin^{-1}\left(\frac{\theta}{2\pi}\right)$

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

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

 Tags angle, circle, cone, relationship

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