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The problem is: The net of a cone is a sector with the angle 160 degrees, and the height (not slant height) of the cone is 2. Find the surface area.

I have already tried setting up the equations

$2^2 + r^2 = s^2$

and

$SA = \pi r^2 + (160/360)(\pi s^2)$

where r = radius of the circle and s = slant height or the radius of the sector of the cone, and SA = surface area

but I can't seem to advance from there. How can I solve this?

For reference, here is what the net of a cone look like:

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2 Answers 2

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If the circle has a radius $r$, then the length of the arc of the sector is $2\pi r$.

The sector has a radius $R$ and its lenght is $2\pi R\cdot \dfrac{160}{360}$ which has to be equal to $2\pi r$. So you have a relation between $R$ and $r$.

Also, if you need to find the relation with the height, you can resamble the cone and you will have a rectangle with the following sides:

    /|
 R / |  h=2
  /__|
   r
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You're missing the fact that the sector has the same arc length as the circle has circumference, so $\frac{160}{360}2\pi s = 2\pi r$. I believe you can figure out everything else from there.

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