# Find the surface area of a cone given height and the angle of the sector in the net

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:

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


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.