Sphere inscribed in a regular tetrahedron

Hello spred

Quote:

Originally Posted by spred

Find the radius of a sphere inscribed in a regular tetrahedron which has a height of 8.

Thank You

The attached diagram shows a vertical cross-section through the tetrahedron, along one of the medians, $AM$ , of the base.

$V$ is the vertex of the tetrahedron, and $O$ the centre of the inscribed sphere.

$G$ is the centroid of the base; $H$ is the centroid of one of the sloping faces. By symmetry, the sphere touches the base and this face at $G$ and $H$ .

The height of the tetrahedron is $h$ . So $VG = AH = h$ . (In your question $h = 8$ .)

Also $GM = HM = \tfrac13AM$ , using the well-known property of the centroid of a triangle. So if $\angle HAM = \theta$ ,
$\sin \theta = \frac{HM}{AM}= \frac13$

$\Rightarrow \tan\theta = \frac{1}{\sqrt8}$ ...(1)

$\Rightarrow GM =HM = h\tan\theta$

$\Rightarrow AG = 2GM=2h\tan\theta$

$\Rightarrow r = OG = AG \tan\theta$
$=2h\tan^2\theta$

$=\tfrac14h$ , from (1)

So the radius of the inscribed sphere is one-quarter of the height of the regular tetrahedron. In your question, then, the radius is $2$ units.

Grandad