I feel that there are two places where there can be confusion in your mind. so, I will try to explain both of them:
1. The word inscribed shape means that shape is completely contained inside and "
fits snugly" inside another geometrical shape.
https://en.wikipedia.org/wiki/Inscribed
2. The concept of radius and height of the cone
Part 1:
Definition & concept of radius: To draw a sphere we choose a point in 3-D and call it center, then we choose a fixed distance and call it the radius. Sphere is then drawn by joining all the points at distance
radius from the center.
So, we can deduce that the distance of all the points on the curved surface of the sphere and the center of the sphere is equal to the
radius.
A hemisphere is a sphere split in half. so, the same rule must apply, thus for a hemisphere also, the distance of all the points on the curved surface and the center is equal to the radius.
Part 2:
Now, height of a right circular cone is: The distance of the top most point (vertex) from the center of the base. Agree?
Part 3:
The circular base of the cone coincides with the circular base of the hemisphere. Thus, if you focus on only the base in a 2-D world, the circles which forms the base of the hemisphere and the cone are the one and the same, thus their centers also coincide.
Part 4:
From all this we can deduce that the vertex of the cone lies on the curved surface of the hemisphere. Thus, height of the cone = radius of hemisphere as their circular base's centers coincide.
Cheers,
Dubes
