The height of the pyramid of minimum volume whose base is a square and whose base triangular faces are all tangents to the sphere.
The sphere has radius r. The base of the pyramid is a regular n-gon.
Let the height of the Pyramid for minimum volume be
Radius of the sphere is
Let the side of the square base be
Area of the square base is
Now from similar triangles rules we get,
Now volume of the pyramid is,
Now we differentiate to get the minimum value,
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