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

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- March 9th 2010, 04:41 PMspredSphere inscribed in a regular tetrahedron
Find the radius of a sphere inscribed in a regular tetrahedron which has a height of 8.

Thank You - March 10th 2010, 05:22 AMGrandad
Hello spredThe attached diagram shows a vertical cross-section through the tetrahedron, along one of the medians, , of the base.

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 ,

...(1)

, from (1)

Grandad