Cells must exchange materials with their surroundings to stay alive. Oxygen, nutrients, wastes, water, and signals usually cross the cell membrane, so the amount of membrane surface matters. As a cell grows larger, its volume increases faster than its surface area.
This makes surface-area-to-volume ratio a major limit on cell size.
Key Facts
- For a cube, surface area = 6s^2, where s is side length.
- For a cube, volume = s^3, where s is side length.
- For a cube, surface-area-to-volume ratio = 6s^2 / s^3 = 6 / s.
- A 1 unit cube has SA = 6 units^2, V = 1 unit^3, and SA:V = 6:1.
- A 2 unit cube has SA = 24 units^2, V = 8 units^3, and SA:V = 3:1.
- A 4 unit cube has SA = 96 units^2, V = 64 units^3, and SA:V = 1.5:1.
Vocabulary
- Surface area
- Surface area is the total area of the outside boundary of an object, such as the cell membrane.
- Volume
- Volume is the amount of space inside an object, such as the cytoplasm inside a cell.
- Surface-area-to-volume ratio
- Surface-area-to-volume ratio compares how much exchange surface an object has for each unit of internal volume.
- Diffusion
- Diffusion is the net movement of particles from an area of higher concentration to an area of lower concentration.
- Cell membrane
- The cell membrane is the thin boundary that controls movement of substances into and out of a cell.
Common Mistakes to Avoid
- Assuming surface area and volume increase at the same rate is wrong because volume grows with the cube of length while surface area grows with the square of length.
- Using SA = s^2 for a cube is wrong because a cube has 6 faces, so its total surface area is SA = 6s^2.
- Thinking a bigger cell always works better is wrong because a larger cell has more cytoplasm to support but relatively less membrane for exchange.
- Ignoring units in the ratio is wrong because surface area is measured in square units and volume in cubic units, so the ratio changes with scale.
Practice Questions
- 1 A cube-shaped cell has side length 3 units. Calculate its surface area, volume, and surface-area-to-volume ratio.
- 2 A cube-shaped cell grows from side length 2 micrometers to 6 micrometers. By what factor do its surface area and volume increase?
- 3 Explain why many small cells can exchange materials more efficiently than one large cell with the same total volume.