# Thread: Finding central angle after slicing cone

1. ## Finding central angle after slicing cone

A cone made of cardboard has a vertical height of 8 cm and a radius of 6 cm. If this cone is cut along the slanted height to make a sector, what is the central angle, in degrees, of the sector?

I drew a diagram. I thought the central angle was 90 degrees because of the 6,8,10 triangle that included the slant height. The answer says 216 degrees.

2. ## Re: Finding central angle after slicing cone

Hello, benny92000!

The problem is not stated clearly.

A cone made of cardboard has a vertical height of 8 cm and a radius of 6 cm.
If this cone is cut along the slanted height to make a sector,
what is the central angle, in degrees, of the sector?

The side view of the cone looks like this:
Code:
              B
*
/:\
/ : \
/  :  \
/   :   \ 10
/    :8   \
/     :     \
/      :      \
A *-------*-------* C
: - 6 - : - 6 - :
We see that the slant height is 10 cm.
The circumference of the base is: .$\displaystyle 2\pi r = 2\pi(6) = 12\pi\text{ cm}$

Now we cut the cone and lay it flat.
We have a large sector of a circle of radius 10.
Code:
                B
* * *
*           *
*               *
*                 *

*         O         *
*         *         *
*        / \        *
10 /   \ 10
*     /     \     *
*   /       \   *
A *           * C
* * *
Length-of-arc Formula: .$\displaystyle s \,=\,r\theta$ . where:.$\displaystyle \begin{Bmatrix}{s &=& \text{length of arc} \\ r &=& \text{radius} \\ \theta &=& \text{central angle}\\[-2mm] && \text{in radians} \end{Bmatrix}$

We have: .$\displaystyle \begin{Bmatrix}s \,=\,\text{arc}(ABC) \,=\, 12\pi \\ r \:=\: 10\end{Bmatrix}$

Therefore: .$\displaystyle 12\pi \,=\,10\theta$

. . . . . . . . . . $\displaystyle \theta \:=\:\frac{12\pi}{10}\text{ radians} \:=\:216^o$

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# what is the formula to calculate height of cone when central angle of sector is given

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