How to Determine a Cube and Sphere of Equal Volume

Three Parts:The TutorialExplanatory Charts, Diagrams, PhotosHelpful Guidance

In this article r1 is used to represent the side of the cube and r2 to represent the radius of the sphere. The formula for the volume V of a cube c is s^3 where s = side (but here r is used for s) so r1^3 = V(c), and the volume of a sphere s is 4/3 πr^3, so in this example 4/3πr2^3 = V(s). The caret symbol, "^", denotes exponentiation for MicroSoft Excel and the article will follow that syntax.

Part 1
The Tutorial

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    Set V(c) = V(s) via r1^3 = 4/3πr2^3
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    r1^3/r2^3 = 4/3π by dividing both sides by r2^3 and simplifying.
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    r1/r2 = (4/3π)^(1/3) = 1.61199195401647 by taking the cube root of both sides and evaluating the right side in Excel as "=(4/3*PI())^(1/3)"
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    Now we can find either r1 or r2 given the other one, for r1 = r2 * 1.61199195401647 and r2 = r1 / 1.61199195401647, where r2 is the radius of the sphere and r1 is the side of the cube.
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    We now have also learned that (4/3π)^(1/3) MEANS the constant of proportion of the volume of a cube equal in volume to a sphere of different basis length r.

Part 2
Explanatory Charts, Diagrams, Photos

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Part 3
Helpful Guidance

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    Make use of helper articles when proceeding through this tutorial:

Tips

  • While it is true that 2πr = circumference C represents "single point multiplication" whereby the area of 2π is swept out while the radius has one point fixed and one point moving in multiplication, and 4s for 4 times a side = the perimeter of a square is "single point translation", that πr2 = Area of circle C is the "bubbling multiplication" of r^2 into a circular area whereas s^2 = the area of a square = s*s is "2 point multiplication" in that one side s is held steady while the other side s transverses its length in multiplicity, that s^3 for a square's volume means the "propagation multiplication" from an origin of a three-dimensional cube, 4/3πr^3 is the "bubbling propagation multiplication" from a point radius of the volume of a sphere being translated from a cube (r^3) by the proportion 4/3π. In other words, there are different sorts of growth -- different sorts of multiplication -- implied by these formulas. And we could also say that in the case of 2πr = circumference C of a circle, that the circumference equals the propagation via a curved radius, or in Excel "=Radians(1)" measuring 0.0174532925199433, of the full distance of 360 degrees = 2π. (In Excel, "=Radians(360)" = 2π, 360/(2*PI())=57.2957795130823 degrees; "=radians(57.2957795130823)" = 1 where 57.2957795130823 is the number of degrees in 1 radian. and 2π * 57.2957795130823 = 360.)

Sources and Citations

  • The source worksheet for this article is "Cube and Sphere wks.xlsx"

Article Info

Categories: Calculating Volume and Area | Mathematics