Inscribed and circumscribed solids
Find the radius of a ball inscribed in a cube or a cone, and of a ball round a cuboid, and the volume of a cylinder round a prism.
- Russia: grade 11
Guide
In the Russian programme combinations of round solids and polyhedra are in Grade 11. A prism inscribed in a cylinder and circumscribed about it is at the advanced level; at the basic level the simple cases (ball and cube, ball and cuboid) are enough.
A ball is inscribed in a polyhedron if it touches all its faces; a ball is circumscribed about a polyhedron if all its vertices lie on the sphere. An axial section or a section through the centre of the ball almost always helps.
The centre of the ball circumscribed about a cuboid is the midpoint of its diagonal, and the radius is half the diagonal. A cylinder can be circumscribed about a right prism if a circle can be drawn round its base; the radius is that circle's radius.
Common slips: giving the diameter instead of the radius, not halving the diagonal, and mixing up the inscribed and circumscribed balls.