Thread: Area of a circle cut out of a cone?

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.

Originally Posted by HyperSquirrel

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.

1. Actually you are asking about the intersection curve (<--- google for it) between a cone and a cylinder.

2. There are a lot of parameters to be considered:
- Radius of the cylinder.
- radius of the base of the cone and
- Height of the cone
- Do the axes of cone and cylinder intersect? If not, how large is the offset?
- How large is the angle included by the axes?

3. I can (vaguely) remember a very hard time before the drawing board when I had to construct the intersection line of 2 cones. You must use auxiliary planes and spheres and the only high-tech equipment we were allowed to use, was a slide-ruler! (This nightmare is now more than 40 years ago, so probably there are a lot of computer programs which will draw the curves)

Thank you for your quick reply. At least we now have an idea what we're dealing with.

Googling the intersection curve turned up a lot of interesting but not particularly helpful things. None of the sites I was able to find actually listed the formulas and/or explained how to calculate it.
In addition there were a bunch of CAD related sites, which would be help from a more practical point of view, but it doesn't help with the mathematical evidence.

Additional info:
-The cylinder is 100mm long at each end, but the sides are unknown. It has an outer radius of 60 and an inner radius of 68.5.
-The (truncated) cone has an outer base radius of 140 and an inner radius of 138.5. It has an outer top radius of 70 and inner radius of 68.5.
-The cone has a height of 180.
-Yes, they intersect.
-There is a 90 degree angle between the surface of the cone and the sides of the cylinder. It's as though you set a can of soda on top of the cone. (New image attached). So I guess if the cone's sides had a 60 degree angle, then the angle between the axes would be 30 degrees? 90 - the angle of the cone's surface relative to the cone's axis?



It's placed on the exact center of the truncated cone.

Do you have handy a formula that takes the parameters you mentioned above? Deriving it would probably be beyond the scope of this course, which is sort of technically-oriented.

Of your circular cylinder is centered on the axis of the cone, and parallel to it, as you seem to be saying, then the "part of the surface of the cone cut out is just a smaller cone. You could, for example, work out where the height at which the cylinder passes throught the cone- the "z" value for which the radius at that height is radius of the circle. The portion cut off, the tip of the cone, has the radius of the circle as base radius and height equal to the height of the original cone minus that "z".

To clarify, the cylinder I'm talking about is the one on top in the image, going into the side of the cone. (Rather than the one that shares an axis with the cone.)

Originally Posted by HyperSquirrel

Thank you for your quick reply. At least we now have an idea what we're dealing with.

....

Do you have handy a formula that takes the parameters you mentioned above? Deriving it would probably be beyond the scope of this course, which is sort of technically-oriented.

Much to my disappointment I have to disappoint you: What you are asking is far beyond my mathematical knowledge and skills. Sorry!

No problem; I appreciate the help.

I consulted with another friend and we decided that the best solution is probably just to approximate it as a circle cut out of a plane. It's an engineering-ish applied question, so it might well be that absent some sort of computer modeling or formula from a book, the intended solution is to approximate.