Surface AreatoVolume Ratio
by Greg Crowther
CHORUS:
Surface area to volume ratio:
If you're small, then it's high;
If you're large, it's low.
Animals aren’t cubeshaped, but let’s pretend.
Say L is the length of a side  what then?
6L to the 2 over L to the 3 equals SA over V.
SA over V.
CHORUS
Animals’ internal metabolic activities can’t
Exceed what they exchange with the environment.
And so, as they get wide and tall,
Their metabolic rates must fall.
Metabolic rates must fall.
CHORUS
Written for Biology 220 at the University of Washington, this song explains the relationship between body size and massspecific metabolic rate. It references the formulas for the surface area and volume of a cube with sides of length L: 6L^{2} and L^{3}, respectively. These formulas should be written out explicitly to avoid confusion. Also, the alliteration of “large” and “low” in the line “If you’re large, it’s low” reminds students to group these two adjectives together (a large body size implies a low surface areatovolume ratio).
Questions: (1) What are some things whose exchange with the environment may be limited by the SA/V ratio? (2) If we were to assume that an animal were spherical, rather than cubeshaped, would SA/V be similarly affected by body size? Answers: (1) Oxygen and heat are the usual answers. Other possibilities include carbon dioxide, nitrogenous waste products like urea, water, etc. (2) Yes. The surface area of a sphere equals 4*pi*r^{2}, where r is the radius. The volume of a sphere equals (4/3)*pi*r^{3}. The surface areatovolume ratio is 3/r, which decreases as r increases. Thus this ratio decreases with increasing size, regardless of whether the object is cubeshaped or spherical.
• MP3 (demo)
• score (with melody playback)
• video
