Surface Area-to-Volume Ratio
by Greg Crowther
Surface area to volume ratio:
If you're small, then it's high;
If you're large, it's low.
Animals aren’t cube-shaped, 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.
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.
Written for Biology 220 at the University of Washington, this song explains the relationship between body size and mass-specific metabolic rate. It references the formulas for the surface area and volume of a cube with sides of length L: 6L2 and L3, 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 area-to-volume 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 cube-shaped, 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*r2, where r is the radius. The volume of a sphere equals (4/3)*pi*r3. The surface area-to-volume ratio is 3/r, which decreases as r increases. Thus this ratio decreases with increasing size, regardless of whether the object is cube-shaped or spherical.
• MP3 (demo)
• score (with melody play-back)