A friend was asking me about this, and I'm not entirely confident in my ability to answer.

If you have a cone, and you cut a circle out of the top, how do you calculate the area of the section removed from the cone? (If you were going to, say, poke a straw into the cone at a right angle.) My calculus is way too rusty for this.

My thinking so far:

So it's trivial to get the area of a circle cut in a sheet.

If you bend the sheet into a cylinder and then cut the hole out, the sheet if straightened back out would have a deformed circle: deformed to make it wider, I guess. Perhaps you'd do something with the size of the cut-out circle, in order to figure out how many degrees it spans, and then ... uh ... get some idea about the extra circumference?

The fact that it's a cone adds another complexity: you're basically doing cuts in a series of increasingly narrow infinitely-thin cylinders. I guess you could do that with an integral, but is there a pre-calculus solution?

Many thanks for any help you can offer.