# Does a circular cone with half-angle $x$ always fit within an elliptical cone of minor half-angle $x$?

I have two right cones. One is a circular cone with height $$h$$ and half-angle $$x$$. EDIT: Both share the same apex and axis.

The other is an elliptical cone, with height $$h$$, minor half-angle $$x$$ and major half-angle $$y$$ where $$y > x$$.

Will the entire volume of the first (circular) cone always fit within the second?

This is for some software, and I need to know if I can make simplifying assumptions about the angles involved. If the circular always fits within the elliptical, than I can do a (much easier) circular cone check in some cases.

Maybe this just simplifies down to whether a circle of radius $$r$$ will always fit within an ellipse of minor axis $$r$$? (If so, then it's true, right?)

Please forgive any mistakes in terminology, this is a bit outside my knowledge base.

• Are you assuming that both cones have the same vertex, and the same axis? Jul 6, 2022 at 15:36
• @DavidSheard I apologize, yes, that is an assumption I'm making. Jul 6, 2022 at 15:48

The answer is yes. One way to see this is to consider the common axis for the two cones, and a point $$p$$ on this axis. Say this point is a distance $$d$$ from the common apex of the cones, and consider the plane $$P$$ orthogonal to the axis at the point $$p$$. This will intersect the two cones in a circle and an ellipse respectively. The circle will have radius $$r=d\tan(x)$$ and will be centred at $$p$$, while the ellipse will have major and minor axes $$d\tan(x)$$ and $$d\tan(y)$$ respectively. Since $$y>x$$, $$d\tan(y)>d\tan(x)$$. This ellipse will also be centred at $$p$$, and so must contain the circle. As this is the case for every point $$p$$ on the axis, the circular cone s contained in the elliptical cone. 