Cell Size Lab – Quick Reference

Concept Overview

  • Purpose: Understand how surface area-to-volume (SA:V) ratios influence exchange of materials between cells and the environment.

  • Key idea: Smaller cells have higher SA:V, enabling more efficient exchange per unit volume.

  • Relevance: SA:V governs uptake of nutrients, waste removal, heat exchange, and overall metabolic exchange.

Simulation & Observations

  • Use the University of Utah Genetic Science Learning Center’s Cell Size and Scale simulation to observe scale differences.

  • Tasks: identify the largest and smallest items and their sizes; identify one eukaryotic cell and its size; identify one prokaryotic cell and its size; compare differences and discuss possible reasons.

SA:V Relationship (conceptual)

  • When size increases, SA:V decreases; when size decreases, SA:V increases.

  • Consequence: Larger cells have more difficulty exchanging materials efficiently across the surface relative to their volume; organisms offset this with folded membranes, microvilli, organelles, and compartmentalization.

Calculations: Shapes and SA:V

  • Formulas are used to compare SA and V for fixed shapes.

Sphere (radius r)

  • Volume: V=43πr3V = \frac{4}{3}\pi r^3

  • Surface Area: SA=4πr2SA = 4\pi r^2

  • SA:V ratio: SAV=3r\frac{SA}{V} = \frac{3}{r}

Sphere example: r = 2\,\text{cm}

  • V=43π(2)3=323π cm3V = \frac{4}{3}\pi (2)^3 = \frac{32}{3}\pi\ \text{cm}^3

  • SA=4π(2)2=16π cm2SA = 4\pi (2)^2 = 16\pi\ \text{cm}^2

  • SAV=16π323π=32 cm−1\frac{SA}{V} = \frac{16\pi}{\frac{32}{3}\pi} = \frac{3}{2} \ \text{cm}^{-1}

Cube (side length s)

  • Volume: V=s3V = s^3

  • Surface Area: SA=6s2SA = 6s^2

  • SA:V ratio: SAV=6s2s3=6s\frac{SA}{V} = \frac{6s^2}{s^3} = \frac{6}{s}

Cube example: s = 3\,\text{cm}

  • V=33=27 cm3V = 3^3 = 27\ \text{cm}^3

  • SA=6×32=54 cm2SA = 6\times 3^2 = 54\ \text{cm}^2

  • SAV=5427=2 cm−1\frac{SA}{V} = \frac{54}{27} = 2\ \text{cm}^{-1}

Rectangular Solid (dimensions a × b × c)

  • Volume: V=abcV = abc

  • Surface Area: SA=2(ab+bc+ca)SA = 2(ab + bc + ca)

  • SA:V ratio: SAV=2(ab+bc+ca)abc\frac{SA}{V} = \frac{2(ab + bc + ca)}{abc}

Note

  • For an arbitrary rectangular solid, use the formulas above; plug in your dimensions to compute SA, V, and SA:V.

Cylinder (radius r, height h)

  • Volume: V=πr2hV = \pi r^2 h

  • Surface Area: SA=2πrh+2πr2=2πr(h+r)SA = 2\pi r h + 2\pi r^2 = 2\pi r(h + r)

  • SA:V ratio: SAV=2πrh+2πr2πr2h=2(h+r)rh\frac{SA}{V} = \frac{2\pi r h + 2\pi r^2}{\pi r^2 h} = \frac{2(h + r)}{r h}

Cylinder example: r = 1\,\text{cm}, h = 3\,\text{cm}

  • V=π(1)2(3)=3π cm3V = \pi (1)^2 (3) = 3\pi\ \text{cm}^3

  • SA=2π(1)(3)+2π(1)2=8π cm2SA = 2\pi (1)(3) + 2\pi (1)^2 = 8\pi\ \text{cm}^2

  • SAV=8π3π=83 cm−1\frac{SA}{V} = \frac{8\pi}{3\pi} = \frac{8}{3} \ \text{cm}^{-1}

Quick Takeaways

  • SA:V controls exchange efficiency; high SA:V favors rapid exchange and thermal regulation.

  • As size increases, SA:V decreases, limiting exchange; cells/organisms adapt with increased surface area (folds, membranes) or internal compartments to maintain functionality.