illustrate the principle that surface area to volume ratios decrease with increasing size by calculating surface areas and volumes of simple 3-D shapes (as shown in the Mathematical requirements)

Movement into and out of Cells

Objective

Illustrate the principle that surface‑area‑to‑volume ratios (SA:V) decrease with increasing size by calculating surface areas and volumes of simple three‑dimensional shapes.

Mathematical Requirements

For each shape we will derive the formulas for surface area (SA) and volume (V), then calculate SA:V for a series of sizes.

1. Sphere

Let \$r\$ be the radius of the sphere.

\$\text{Surface area: } SA_{\text{sphere}} = 4\pi r^{2}\$

\$\text{Volume: } V_{\text{sphere}} = \frac{4}{3}\pi r^{3}\$

Thus the ratio

\$\frac{SA}{V} = \frac{4\pi r^{2}}{\frac{4}{3}\pi r^{3}} = \frac{3}{r}\$

The SA:V is inversely proportional to the radius.

2. Cube

Let \$a\$ be the length of one edge.

\$\text{Surface area: } SA_{\text{cube}} = 6a^{2}\$

\$\text{Volume: } V_{\text{cube}} = a^{3}\$

Ratio

\$\frac{SA}{V} = \frac{6a^{2}}{a^{3}} = \frac{6}{a}\$

Again, SA:V decreases as the edge length increases.

3. Cylinder (closed at both ends)

Let \$r\$ be the radius of the base and \$h\$ the height.

\$\text{Surface area: } SA_{\text{cyl}} = 2\pi r^{2} + 2\pi r h\$

\$\text{Volume: } V_{\text{cyl}} = \pi r^{2} h\$

Ratio

\$\$\frac{SA}{V} = \frac{2\pi r^{2} + 2\pi r h}{\pi r^{2} h}

= \frac{2}{h} + \frac{2}{r}\$\$

For a cylinder where \$h = 2r\$ (a common proportion), the ratio simplifies to \$\frac{3}{r}\$, showing the same inverse relationship.

Numerical Comparison

Below are calculated SA, V and SA:V for each shape at three representative sizes.

Interpretation

• For each shape, the SA:V ratio is inversely proportional to the characteristic linear dimension (radius or edge length).
• Doubling the size roughly halves the SA:V ratio.
• Smaller cells have a larger SA relative to their volume, allowing more efficient exchange of nutrients and waste across the plasma membrane.
• As cells grow, the decreasing SA:V ratio limits diffusion rates, which is why many cells adopt strategies (e.g., flattening, forming microvilli, or becoming multinucleated) to maintain a favourable surface area.

Biological Relevance

In living organisms, the principle explains why:

1. Most animal cells are small (typically 10–30 µm in diameter).
2. Plant cells develop large vacuoles that push the cytoplasm into a thin layer, effectively increasing surface area relative to volume.
3. Specialised structures such as alveoli in lungs or villi in intestines dramatically increase the effective surface area for gas exchange and nutrient absorption.

Summary

The calculations above demonstrate mathematically that as a cell (or any 3‑D object) becomes larger, its surface‑area‑to‑volume ratio falls. This geometric constraint underpins many cellular adaptations that maximise exchange with the environment.