surface area to volume relationship

drawing of a cube

Imagine a cube 1 foot on a side (substitute “meter” for “foot” if you like).  Its volume will be 1 cubic foot. Each side is one square foot, so its total surface area is 6 square feet (because the cube has 6 sides).


cube 3 feet on a side

Now imagine a cube 3 feet on a side. Its volume will be 27 cubic feet (3 × 3 × 3). Each side is 9 square feet (3 feet by 3 feet), so its total surface area is 54 square feet.

What has happened? The linear dimensions are 3 times bigger. The surface area has increased 9 times (54 ÷ 6). But the volume has increased 27 times (27 ÷ 1).  

Let's calculate surface area to volume ratios:

For the 1-foot cube, 6:1.

For the 3-foot cube,  54:27, or 2:1.

For a 10-foot cube, 600:1000, or 0.6:1.

As the cube gets bigger, the volume increases much more rapidly than the surface area, because the volume increases as the cube of the linear dimension, but the surface area increases as the square. This relationship applies, not just to cubes, but to spheres and any other fixed shape.


Some examples of the importance of this relationship

The ratio of surface area to volume of a baby is much greater than that of an adult. Heat production is more or less proportional to volume. Heat loss and gain is proportional to surface area. As a result, in unfavorable temperatures a baby will become distressed much more rapidly than an adult. 

An ant, like other insects, has an exoskeleton. The rigid portion of its body, which gives the ant its shape, is on the outside. Imagine an ant doubling in size. The exoskeleton, which is surface area, increases as the square and will be 4 times bigger, but the weight of the ant will increase as the cube.  It will be 8 times heavier. At some point the surface area:volume ratio reaches a point at which the exoskeleton is no longer able to support the increased weight. You will never see elephant-sized ant.

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