is the vertex of the tetrahedron, and the centre of the inscribed sphere.
is the centroid of the base; is the centroid of one of the sloping faces. By symmetry, the sphere touches the base and this face at and .
The height of the tetrahedron is . So . (In your question .)
Also , using the well-known property of the centroid of a triangle. So if ,So the radius of the inscribed sphere is one-quarter of the height of the regular tetrahedron. In your question, then, the radius is units.
, from (1)