Calculate surface-area-to-volume ratio
For geometrically similar objects, surface area grows with length squared and volume with length cubed, so SA:V falls as size increases.
AS Level · Cambridge International Biology 9700, 2028–2030. These lessons explain membrane transport and investigation reasoning. Written practice is self-checked; it does not demonstrate supervised practical competence or replace teacher-reviewed assessment.
Before you begin
- Relate phospholipid and protein structure to their roles, and distinguish a concentration gradient from the direction of net movement.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
When all dimensions of a similar object double, by what factor does surface area change?
What you will learn
- Calculate surface area, volume and their ratio for simple three-dimensional shapes
- Explain why surface-area-to-volume ratio decreases as a similar object becomes larger
Calculate before comparing
For a cube of side length a, surface area is 6a² and volume is a³. Surface-area-to-volume ratio is 6a²:a³, or 6/a when expressed as a numerical ratio in consistent length units.
Use compatible units and report which ratio convention is used. Surface area has squared units and volume cubed units; their ratio has inverse-length units unless expressed as a ratio of the numerical values in the same unit system.
Scaling explains the pattern
If all linear dimensions are multiplied by a factor k, surface area is multiplied by k² and volume by k³. Therefore SA:V is multiplied by 1/k and decreases as a geometrically similar object becomes larger.
A higher SA:V gives more exchange surface relative to internal volume in a model. This geometric principle helps explain diffusion limits, but it does not by itself establish a complete mechanism or organismal adaptation.
Pause and explain
When a similar object becomes larger, what happens to its SA:V ratio?
Worked example
A cube has side 2 cm. Calculate surface area, volume and the numerical SA:V ratio. Compare it with a cube of side 4 cm.
Show the worked solution
- For side 2: SA = 6×2² = 24 cm² and V = 8 cm³.
- SA:V = 24:8 = 3:1, or 3 cm⁻¹.
- For side 4: SA = 96 cm² and V = 64 cm³, ratio 1.5:1.
Answer Side 2 cm: surface area = 24 cm², volume = 8 cm³ and SA:V = 3:1 (3 cm⁻¹). Side 4 cm: surface area = 96 cm², volume = 64 cm³ and SA:V = 1.5:1 (1.5 cm⁻¹). Doubling linear size halves SA:V.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Calculate SA:V for a cube of side 3 cm.
Give me a hint
Use SA=6a² and V=a³.
Compare my reasoning
- SA=6×9=54 cm².
- V=27 cm³; ratio 54:27=2:1.
Surface area = 54 cm² and volume = 27 cm³. SA:V = 54:27 = 2:1, or 2 cm⁻¹ using these units.
Look for these in your work
- Both quantities correct
- Ratio simplified
A cube’s side length increases from 1 cm to 5 cm. State the factors by which its surface area and volume increase.
Give me a hint
Use the square and cube of the linear scale factor.
Compare my reasoning
- Linear scale factor is 5.
- Surface area factor 5²=25; volume factor 5³=125.
Surface area increases by a factor of 25 and volume by a factor of 125.
Look for these in your work
- Surface area factor correct
- Volume factor correct
A spherical cell model has the same shape but twice the radius of another. Predict the relative change in SA:V without calculating exact values.
Give me a hint
Compare linear scaling of area and volume.
Compare my reasoning
- Area scales by 2²=4.
- Volume scales by 2³=8; SA:V halves.
The larger sphere has four times the surface area and eight times the volume, so its SA:V ratio is half that of the smaller sphere.
Look for these in your work
- Area and volume scaling correct
- Ratio change inferred
Common mistakes
- Multiplying surface area and volume by the same scale factor
- Using length units inconsistently in the ratio
- Claiming SA:V alone proves an organism’s entire exchange strategy
What can you explain now?
When all dimensions of a similar object double, by what factor does surface area change?
Compare with the explanation
By a factor of four
Surface area scales with length squared.
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Make it stick
How do surface area and volume scale with linear size, and why does their ratio decrease in larger similar shapes?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.