The Calippo™ popsicle has a specific shape, that I would describe as a circle of radius $r$ and a line segment $l$, typically of length $2r$, that's at a distance $h$ from the circle, parallel to the plane the circle is on, with its midpoint on a perpendicular line that goes through the centre of the circle.
The shape itself consists of lines connecting the circle to line segment.
Do I need to specify how the points on the circle map to a point on the line segment?
Obviously, the points on the circle directly under the end points of the line segment should map "straight up" to those end points. The points on the circle half way between those, at $\tfrac{\pi}{2}$, should map to the mid point of the line segment.
But intuitively there should be a mapping that gives the "most outer" shape such that even if every point on the circle is connected to every point on the line segment, those lines never leave "the popsicle".
Thanks to a comment by Mark S. I now know this is called the convex hull of the circle and the line segment.Given this description, how do I calculate the surface and the volume of this shape?
Does this shape have an official name?
It's not the round chisel as shown in this answer, since it lacks the edge in the shape of half an ellipse.