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Will a sphere that has the identical surface area as a cube fit inside the cube?

The surface area of a cube is 6L^2 where L is the length of a side. The surface area of a sphere is 4 pi r^2 where r is the radius of the sphere

Setting these two terms equal (which we do because the surface areas are equal, [we do not need and exact number, mind you. We just have to know they are equal.]) and doing some algebra we find that the cube has a side 1.414 times as long as the radius of the sphere.

BUT WAIT. The radius of the sphere is only half its size (diameter)! To hold the sphere, the cube needs a side of 2r or twice the sphere's radius!! It is only 1.414 times as long so, the sphere will not fit.

Robert Avakian

A sphere and a cube cannot have the "identical" surface area. The surface area of a cube is: Scube = 6 x L^2 where L is the length of the edge -- all edges being equal. The surface area of a sphere: Ssphere = 4 x pi x (L/2)^2 where (L/2) is the radius of the sphere. Since pi = 3.1415926... is a transcendental number, its decimal expression never terminates. As a result Ssphere can never exactly equal Scube.

Vince Calder

No, the sphere will not fit inside the cube. The cube will not fit inside the sphere, either.

The surface area of a cube whose sides have length a is {6 a^2}. Its longest diagonal is {a sqrt(3)} = 1.732 a.

The surface area of a sphere whose radius is r is {4 pi r^2}. We can find the size of a sphere with the same surface area as a cube by setting these surface areas equal and solving for the radius of the sphere.

4 pi r^2 = 6 a^2
r^2 = 6 a^2 / (4 pi)
r = a sqrt(3/(2 pi))
r = a (0.691)

For the sphere to fit inside the cube, its diameter 2r would have to be equal to or less than the length of one edge of the cube. It is not. The diameter of the sphere is 1.382 times as great as the width of the cube.

You might then wonder if perhaps the cube would nestle inside the sphere. It turns out this will not happen, either: the diagonal of the cube is 1.732 a, which is greater than the diameter of the sphere. So if the cube and sphere were placed atop each other, the sphere would project beyond the cube and the cube would project beyond the sphere.

You did not ask about the volumes of the sphere and the cube. It turns out that the sphere is the shape enclosing the greatest volume inside the least surface area. The volume of a cube of length a on a side is simply a^3. The volume of a sphere with radius r is {4 pi r^3 /3}. In this case, with r = {a sqrt(3/(2 pi))}, the volume of the sphere is {a^3 sqrt(6/pi)} = 1.382 a^3.

So, interestingly, just as the diameter of the sphere is 1.382 times the width of the cube, the volume of the sphere is 1.382 times the volume of the cube. Unfortunately, I cannot find anything more than coincidence in that. I guess I will need to leave it to somebody smarter than me to tell if it means anything.

Richard Barrans, Ph.D., M.Ed.
Department of Physics and Astronomy
University of Wyoming

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