Cone with a spherical cap

Hello, I am new to this forum and a beginner with OpenGL. I am coding in Java. Our current code base includes rendering for primitives such as cones and spheres. However, I need to make something a bit more complex: a cone with a spherical cap, or spherical cone. This is essentially a cone within a sphere, with the cone and sphere sharing the same center point. I need to render the cone plus the spherical cap formed where the base circle of the cone touches the sphere. The best description I could find for this type of shape is here:

Any thoughts on how I can draw this?

Thank you for your help,