Q & A: How to find Surface area and Volume Ratio

how do you find the surface area to volume ratio Thanks’ Kyle
- kyle (age 14)
ohio

Surface area to volume ratio can be found easily for several simple shapes, like for example a cube or a sphere.

For a cube, the equation for surface area is S=6*L*L, where L is the length of a side. Similarly, the volume of a cube is V =L*L*L. So for a cube, the ratio of surface area to volume is given by the ratio of these equations: S/V = 6/L.

For a sphere, surface area is S= 4*Pi*R*R, where R is the radius of the sphere and Pi is 3.1415... The volume of a sphere is V= 4*Pi*R*R*R/3. So for a sphere, the ratio of surface area to volume is given by: S/V = 3/R.

For other shapes you may be able to look up the equations for surface area and for volume, or you may be able to use surface integrals and volume integrals to calculate them yourself (if you've taken calculus, that is.)

One interesting thing that you should notice is that the surface area to volume ratio is inversely proportional to the size of an object. For example, we found that for a sphere S/V is 3/R. Since R tells us the "size" we see that S/V is inversely proportional to this (in other words, as R gets small the S/V ratio gets big). This trend is true for any shape, and is the reason why icing sugar dissolves faster in water than regular sugar, why water evaporates faster if you spray it as a fine mist than if you leave it in a bucket, and why dusty grain elevators sometimes explode if someone lights a match inside (see if you can figure these examples out).

-Kim- (and Tamara & Mats)

(published on 10/22/2007)