How to Determine a Cube and Sphere of Equal Volume

Last Updated: March 4, 2021

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

Part 1 of 2:
The Tutorial

1
Set V(c) = V(s) via r1^3 = 4/3πr2^3
2
r1^3/r2^3 = 4/3π by dividing both sides by r2^3 and simplifying.

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3
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)"
4
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.
5
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.

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Part 2

Part 2 of 2:
Helpful Guidance

1
Make use of helper articles when proceeding through this tutorial:
- See the article How to Determine a Square and Circle of Equal Perimeter for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.
- For more art charts and graphs, you might also want to click on Category:Microsoft Excel Imagery, Category:Mathematics, Category:Spreadsheets or Category:Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.