Question

Question #54a35

Answer 1

We know that each edge of a regular tetrahedron is the diagonal of a surface of a cube whose 8 corner points lie on the sphere circumscribing the tetrahedron.

We are to determine the surface area and volume of that sphere.
We have been given the length of the edge of tetrahdron as `10 " in"`

Now if edge of the cube be x inch,then the length of the diagonal of its any square face will be `sqrt2x" in"`

So by the condition
`sqrt2x=10=>x=5sqrt2" in"`

Now diagonal of the cube
`sqrt3x=sqrt3xx5sqrt2=5sqrt6" in"`
This diagonal will be diameter of the sphere circumscribing the tetrahedron.

So the radius of the sphere will be `r=5/2sqrt6" in"`

So surface area of the sphere
`=4pir^2=4*3.14*(5/2sqrt6)~~471" sqin"`

And volume of the sphere
`=4/3pir^3=4/3*3.14*(5/2sqrt6)^3~~962" cuin"`