Surface Area of a Cone

So I recently encountered a challenging math problem intended for an 8th grader. Find the surface area of a frustum. That is, a circus elephant stand or a bucket. Given the circumference of the top and bottom and the height between.

Wow, that's a tough problem for an 8th grader and I am frustrated at this frustum problem because I never learned the formulas needed. It would be relatively straight forward if I know how to find the surface area of a cone but I don't know. Not that I forgot it but just never learned it, although I know how to find the volume of a cone. That 1/3 base * height formula is actually not exactly straight forward to derive either. I know these formulas are not so useful in life, I know. But the spirit of math should be: if there is an answer find the answer.

Nowadays it is easy to look up the formula. Note the cone has the side and the bottom. The formula looks like this:

That looks a little intimidating. A cone is a paper cup. Surface area is the paper cup plus the bottom circle. The area of the paper cup is the "lateral surface area" and it is π r L, where r is the radius, and L is the slanted height. The slanted height L is also square root of h² + r², straight from the Pythagorean Theorem. Neat. So the formula above is area of the paper cup plus the area of the bottom circle. But where did this lateral area formula π r L come from. I looked up a few explanations and none provide satisfactory explanation. Allow me explain.

Now look at the paper cup. Imagine flattening it. It is a section of a circle. What is its radius? L.
What's the circumference of the big circle that the paper cup is part of? 2 π L.
What's its area? π L ².
But we are interested in the section, the paper cup. Geez I'll need the angle but I don't know.
But I do know its rim's length. That would be the circumference of the bottom of the cone: π r².
Then the paper cup's area is a fraction of the big circle's area π L ².
The fraction is rim over entire circumference, which is π r² over 2 π L. Multiply it out, many things cancel, and then you will get π r L. See, if the cone is entirely flat, then r=L it is just π r².

Armed with this lateral cone area formula π r L, there is still a bit of calculation to find surface area of a frustum, which is top and bottom circle area. Plus the difference of lateral area of bigger cone and smaller cone. First I'll need the height of the cone which takes a bit of work with similar triangles to find out.

But the point is this: Do not ever just tell kids to use formulas, derive them! Deriving formula is learning math. Using the formula is applying math. Plugging in numbers in a formula which you don't know where it came is lowest form of math.