<span>199 square feet.
This problem could be worked out two ways. First is to use the radius and slant height to calculate the height, then use the formula for the surface area of a cone. The other is to work it out from basic principles. I'll solve this from basic principles.
You can consider the cone to be made of two surfaces. The base whose area is pi*r^2 and the cone part on top. If you imagine cutting a straight line from the base to the tip of the cone, you can flatten it into a circle with a wedge cut out of it. The side of that partial circle will have the same length of the curve as the perimeter of the base. So you can use that to calculate that area as well.
Area of top without wedge cut out is:
pi*6^2 = 36 pi.
Perimeter of top without wedge cut out is:
2*pi*6 = 12 pi
Perimeter of base is:
2*pi*5.5 = 11 pi.
So the area of the top is 11pi/12pi * pi * 6^2 = 11/12 * pi * 36 = 33pi.
And the area of the base is pi*r^2 = pi*5.5^2 = 30.25pi.
So the overall area of the cone is:
33pi + 30.25pi = 63.25pi = 198.7057353 = 199.
To show that the above method was correct, let's do it the conventional way. We can use the Pythagorean theorem to get the height. So:
sqrt(6^2 - 5.5^2) = sqrt(36 - 30.25) = sqrt(5.75) = 2.397915762
And the surface area of a cone is:
A=pi*r(r+sqrt(h2+r2))
Substitute the known values. And since we know the square already, I'll use that more precise value:
A=pi*5.5(5.5+sqrt(5.75+30.25))
A=pi*5.5(5.5+sqrt(36))
A=pi*5.5(5.5+6)
A=pi*5.5(11.5)
A=pi*63.25
A=198.7057353
And as you can see, you get the exact same answer.</span>
