How this calculator works
Surface area grows with the square of length while volume grows with the cube. Dividing area by volume therefore leaves one factor of length underneath, giving units of reciprocal length. A small object has a high ratio and a large object of the same shape has a low ratio, which is why small animals lose heat quickly and why powders dissolve faster than blocks. A sphere always has the lowest ratio for a given volume, and a long thin shape has a high one.
Formula
Sphere: A/V = 3/r. Cube: A/V = 6/a. Cylinder: A/V = 2(r + h)/(rh). Rectangular prism: A/V = 2(lw + lh + wh)/(lwh).
Worked example
A sphere of radius 3 m has area 36π ≈ 113.097 m² and volume 36π ≈ 113.097 m³, so its ratio is 1.0 per metre. A cube of side 4 m has area 96 m² and volume 64 m³, a ratio of 1.5 per metre, and a 4 m by 3 m by 2 m prism has a ratio of 52/24 ≈ 2.167 per metre.
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Frequently asked questions
What units does the ratio use?
The ratio is surface area divided by volume, so it is in reciprocal metres. Enter measurements in any supported unit and the ratio is still reported per metre, because the engine converts to metres first.
Why does the ratio fall as size increases?
Doubling every length multiplies the surface area by four but the volume by eight. The ratio therefore halves, even though the shape is unchanged. This is the square-cube law.
Which shape is most efficient?
A sphere has the smallest surface area for a given volume and therefore the lowest ratio. Among the shapes here, a sphere at radius 3 has a ratio of 1.0 while a cube of side 3 would have a ratio of 2.0.