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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.
Steps
The Tutorial
- Set V(c) = V(s) via r1^3 = 4/3πr2^3
- r1^3/r2^3 = 4/3π by dividing both sides by r2^3 and simplifying.Advertisement
- 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)"
- 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.
- 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.Advertisement
Expert Q&A
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.)Thanks
References
- The source worksheet for this article is "Cube and Sphere wks.xlsx"
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