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Math Help - Sphere inscribed in a regular tetrahedron

  1. #1
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    Sphere inscribed in a regular tetrahedron

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

    Thank You
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  2. #2
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    Hello spred
    Quote Originally Posted by spred View Post
    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
    Attached Thumbnails Attached Thumbnails Sphere inscribed in a regular tetrahedron-untitled.jpg  
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