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Bill1939's avatar

Am I doing something dumb with arithmetic?

Asked by Bill1939 (6070 points ) January 25th, 2013

Area of a sphere = 4 times Pi times radius squared.
Area of a cube = 6 times length squared.

The area of a cube with the volume of one cubed is equal to 6; however the area of a sphere with that volume is 4.8359? I would have assumed that the area of a volume be the same regardless of its shape.

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13 Answers

SadieMartinPaul's avatar

You have 36 meters of fence, which you want to use to enclose a garden.

You decide to make a long, skinny garden. It’s 17 m long and 1 m wide. The area of your garden is 17×1 = 17 m sq.

Instead, you change your mind and create a square garden, 9 m on each side. The area of your garden is 9×9 = 81 m sq.

LostInParadise's avatar

A sphere is the most compact shape, meaning that for a given area it will have the smallest volume. That is why water drops assume a spherical shape. Surface tension forces the water to assume the shape with the smallest volume.

Mariah's avatar

You’re talking about surface area, right?

The answers above have got it. If you make a shape long and skinny, it’s intuitive that more of its volume is at the surface than in the middle. If you squish it in, the opposite is true. A sphere is ultimate in “squished” shapes.

zenvelo's avatar

Think of it this way- a sphere with diameter equal to the length of one edge of the cube will “fit” inside the cube. (Actually, there will be a single tangential point on each of the six sides of the cubes in the center of each side). The rounded “corners” are missing from the sphere. So the volume and surface area of the sphere is less than that of a cube.

Its the same reason a circle of diameter 1 fits inside a 1×1 square.

CWOTUS's avatar

I was on my way to saying almost exactly the same thing @zenvelo has just said.

In terms of actual units, the cube of 1 cubic foot (to use a unit) would have sides of 1 foot each.

The sphere of 1 cubic foot has a diameter of 1.24 feet. It won’t fit into the 1^3 cube. The sphere therefore does not “use space more efficiently”, it just uses its surface more efficiently to enclose the given volume.

Bill1939's avatar

@zenvelo, if a sphere with 1 cubic volume was centered inside a cube with 1 cubic volume, the sphere would penetrate all six sides of the cube (by 0.12 at each of the diameter’s ends) while not reaching the corners. If the diameter was equal to the length of the cube’s side, then the sphere’s area would be 0.3.14159 and the volume 0.52360.

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Bill1939's avatar

In trying to rap my head around a Planck scaled universe, I discovered the area difference between cubic and spherical volume. I’m told that Planck volume is seldom used in calculations (unfortunately my tables were based upon volume), Planck area being preferred. Math skill limits my ability to determine whether the area of a quantum is 6 or 4.84 square Planck lengths, whether the area defines a cube or a sphere and, if it is a sphere whether Planck area or Planck volume defines the quanta.

Yetanotheruser's avatar

I’ll bet it’s easier than wrapping your head around an Escher scaled universe! Or maybe not…

Bill1939's avatar

The two may not be so different, @Yetanotheruser.

gasman's avatar

Surface area-to-volume ratio (A / V) determines many physical properties of solids & how they interact with their environment. As others explained above, this ratio is minimized for a sphere & equals
A/V = (4πr^2) / [(4/3)πr^3] = 3/r.
For a cube it equals
A/V = 6r^2 / r^3 = 6/r.
Already twice that for a sphere.

There is, in fact, no upper limit on this ratio. In a variety of fractal solids (in the true sense of infinitely recursive iterations at ever-smaller scales) a finite volume may be bounded by an infinite surface area. By extension to the real world, that’s how one cubic centimeter of a material like activated charcoal can have an absorptive surface area equal to several tennis courts! Likewise many biological activities are highly dependent on this ratio: lungs, blood vessels, etc. are highly convoluted, near-fractal structures in which membrane-bound processes (such as gas exchange) operate efficiently because of huge A/V ratios.

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gasman's avatar

Good grief, talk of doing something dumb… I think my last paragraph might be ok but the rest was nonsensical & “not even wrong,” to quote Pauli. I plead temporary insanity. My new calculations agree with @CWOTUS‘s posting long ago: The sphere of 1 cubic foot has a diameter of 1.24 feet. It’s surface area is found to be 4.835 square feet, compared to 6 square feet for a one-foot cube. Thus at equal volumes a cube’s area is 24% larger than a sphere’s (not double).

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