How does volume relate to surface area?

Question

Math homework and i dont know

Answer 1

I will just expand on previous answerer. So if you have a cube 1x1x1 the volume is 1 cubic inch and surface 6 square inches. Now make a rectangle box by making one side rectangle ax(1/a) to keep volume the same. Your volume remains the same 1xax(1/a) = 1 cubic inch. However, your surface is 2*1*a + 2*1*(1/a) + 2*a*(1/a) = 2*(1 + a + 1/a). Then for a = 1/4, 1/2, 2/3, 1, 3/2, 2, 3, your surface is 10.5, 7, 6.333, 6, 6.333, 7, 10.5 respectively. This proves that out of all boxes the cube has the smallest surface for the given volume.
Similarly for the egg like shapes the sphere has the smallest surface/volume ratio. Etc.

Answer 2

Volume and surface Area - Subject = Cubes

I can discuss cubes. somehow I have forgotten the formula
for surface area of a sphere, so I will deal with cubes only.

First example: Select a cube with sides of 1 inch each.
The Volume of this cube is 1 x 1 x 1 inch = 1 cubic inch.
The surface Area is 6 x (1 x 1 inch) = 6 square inches
( a cube has six sides, all 1 inch by 1 inch in the example.)

So, in the example chosen, for a 1 x 1 x 1 inch cube, the
volume is 1 cubic inch, and the surface area is 6 square inches.

If you do the math, a 2 x 2 x 2 inch cube would have 8 cubic inches of volume and 24 square inches of surface area.

Again, if you do the math, a 3 x 3 x 3 inch cube would have
27 cubic inches of volume, and 54 square inches of surface area.

And a 4 x 4 x 4 inch cube has 96 square inches of surface area, and 64 cubic inches of volume.

Etc., etc., etc.

Were you to squash the cube into a very thin rectanglular mass such as a chunk of plywood, the surface area would go way up and the volume would go way down. You might continue squashing until the rectangular mass had almost no thickness at all - such as in a huge slice of paper. In that case, the surface area has almost been maximized, and the volume has been reduced to almost nothing.

Answer 3

This talks about organisms, but it shows the volume-surface area relationship:
http://www.tiem.utk.edu/~gross/bioed/bealsmodules/area_volume.html

Notice that for any increase, x * l or x * r, in length or radius, the increase in surface area is x squared (x^2) and the increase in volume is x cubed (x^3). For example, when length is doubled (i.e., x = 2) surface area is quadrupled (22 = 4) not doubled, and volume is octupled (23 = 8) not tripled.

Answer 4

Ughh, I'm doing the same thing in math.
I hate it. I hate math in general.

The answer would be no.

Answer 5

the integral formulas of stokes tell you the relationship.

see for instance http://cnx.org/content/m12866/latest/

Answer 6

What kind of geometric objects are you interested in?