What are some examples of surface area to volume ratio?

Question

Maya Anderson · Accepted Answer

The surface-area-to-volume ratio or SA:V, is the amount of surface area of an organism divided by its volume.

Assume that you are a spherical cell.

Your SA:V is important because you depend on diffusion through your cell wall to obtain oxygen, water, and food and get rid of carbon dioxide and waste materials.

Let's calculate SA:V for three cell sizes.

#"SA" = 4πr^2# and #V = 4/3πr^3#

r = 1 mm: #SA = 4π " mm"^2; V= 4/3π " mm"^3; "SA:V" = 3.0#

r = 2 mm: #SA = 16π " mm"^2; V= 32/3π " mm"^3; "SA:V" = 1.5#

r = 3 mm: #SA = 36π " mm"^2; V= 108/3π " mm"^3; "SA:V" = 1.0#

Your surface area to volume ratio decreases as you get bigger.

Now let's assume that nutrients can diffuse into your cell at the rate of 0.05 mm/min. In 10 min they would reach 0.5 mm to the centre. What fraction of your cell would still be unfed after 10 min?

r = 1 mm

#V_"tot" = 4/3π " mm"^3#

#r_"unfed" = "0.5 mm"#

#V_"unfed" = 4/3πr^3 = 4/3π×(0.50" mm")^3 = 0.50/3π" mm"^3#

#% "unfed" = V_"unfed"/V_"tot" × 100 % = (0.50/cancel(3) cancel("π mm³"))/(4/cancel(3) cancel("π mm³")) × 100 % = 12 %#

r = 2 mm

#V_"tot" = 32/3π" mm"^3#

#r_"unfed" = "1.5 mm"#

#V_"unfed" = 4/3πr^3 = 4/3π × ("1.5 mm")^3 = 13.5/3π" mm"^3#

#% "unfed" = V_"unfed"/V_"tot" × 100 % = (13.5/cancel(3) cancel("π mm³"))/(32/cancel(3) cancel("π mm³")) × 100 % = 42 %#

r = 3 mm

#V_"tot" = 108/3π" mm"^3#

#r_"unfed" = "1.5 mm"#

#V_"unfed" = 4/3πr^3 = 4/3π×("2.5 mm")^3 = 62.5/3π" mm"^3#

#% "unfed" = V_"unfed"/V_"tot" × 100 %(62.5/cancel(3) cancel("π mm³"))/(108/cancel(3) cancel("π mm³")) × 100 % = 58 %#

The bigger you get, the longer it takes for the nutrients to reach your interior.

Beyond a certain limit, not enough nutrients will be able to cross the membrane fast enough to accommodate your increased volume.

You will have to stop growing if you want to survive.

Nicholas Ahn · Answer

undefined A high surface area to volume ratio is advantageous for many applications, including drug delivery, electronics, and chemistry, where catalysts are used to increase the frequency of interactions between reactants and active sites. In engineering, heat exchangers and cooling systems are designed to maximize surface area to volume ratios to enhance heat transfer efficiency. Examples of surface area to volume ratios include small organisms, such as bacteria or cells, where a large surface area relative to volume facilitates efficient exchange of nutrients and waste products.