Surface Area to Volume Ratio

Essential Question

Why are most cells so small?

Most cells are far smaller than the period at the end of this sentence. In this lesson, you’ll calculate the surface area to volume ratio of cubes, see how the ratio changes as size increases, and use it to explain why cells stay small and how some cells increase their surface area.

Surface Area and Volume

• Surface area (SA) is the area of the outside
• Volume (V) is the space inside
• For a cube: SA = 6s² and V = s³
• Divide SA by V to get the ratio

Surface area (SA) is the total area of the outside of an object, and volume (V) is the space inside it. For a cube with side s, the cube has 6 faces, so SA = 6 x s x s = 6s², and V = s x s x s = s³. The surface area to volume ratio (SA : V) is the surface area divided by the volume. For a cube, SA : V = 6s² / s³ = 6 / s. A cell is not a cube, but cubes make the pattern easy to calculate.

The Ratio Falls as Size Grows

• Surface area grows with the square of the side
• Volume grows with the cube of the side
• So volume grows faster than surface area
• SA : V falls as the object gets bigger

When the side of a cube doubles, its surface area becomes 4 times larger (2²), but its volume becomes 8 times larger (2³). Volume increases faster than surface area, so the ratio SA : V falls as the object gets bigger. The same is true of other shapes. For a sphere of radius r, the ratio is 3 / r, which also falls as the radius grows.

Materials Must Reach the Whole Cell

• Materials enter through the surface
• They must then reach all of the volume
• In a larger cube, less of the volume is near the surface
• A cell that is too large is not served well

A cell takes in nutrients and gets rid of wastes through its membrane, which is its surface. Those materials must then spread through the whole volume of the cell. Agar cubes soaked in dye model this: when the dye reaches 0.5 cm in from every face, it reaches all of a 1 cm cube but only part of a 4 cm cube. As a cell grows, its volume (which needs supplies) increases faster than its surface (which brings them in).

Diffusion Distance

• Diffusion is fast over short distances
• It is slow over long distances
• Time grows with the square of the distance
• Small cells keep every part close to the surface

Materials move through the cytoplasm mostly by diffusion, and diffusion gets slow as the distance grows. In a simple model, the time to diffuse is the distance squared divided by 2D, where D depends on the substance. D = 1,000 µm²/s is typical for a small molecule in water; inside a real cell, diffusion is slower. Ten times the distance takes 100 times as long. A cell 10 µm across can spread a molecule in a fraction of a second, but spreading it over 1,000 µm would take minutes. Small size keeps every part of the cell close to the surface.

Ways to Increase Surface Area

• Microvilli fold the membrane of intestinal cells
• Mitochondria fold their inner membrane
• Root hair cells extend long projections
• More surface for the same volume

Some cells and organelles increase their surface area without becoming much larger. Intestinal lining cells have microvilli, tiny folds of membrane that increase the surface for absorbing food. A mitochondrion folds its inner membrane, and a root hair cell has a long projection. Each of these adds surface area, so more of the cell is in contact with the outside. A cell that grows too large cannot be served well by its surface, which is one reason most cells stay small. You will see in Unit 7 that cells divide instead of growing without limit.

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Worked Examples

Try each problem on your own first, then open the solution to check your work.

Example 1: Calculate the SA : V Ratio

Model two cells as cubes. Cell A has a side of 5 µm, and Cell B has a side of 10 µm. Find the surface area, volume, and SA : V ratio of each. Which cell has the larger ratio?

Cell	Side (µm)
A	5
B	10

Show the solution

1. Cell A: SA = 6 x (5 µm)² = 6 x 25 µm² = 150 µm². V = (5 µm)³ = 125 µm³. SA : V = 150 / 125 = 1.2 : 1.
2. Cell B: SA = 6 x (10 µm)² = 600 µm². V = (10 µm)³ = 1,000 µm³. SA : V = 600 / 1,000 = 0.6 : 1.
3. Compare: 1.2 is greater than 0.6, so the smaller cell has the larger ratio.
4. Answer: Cell A: 1.2 : 1. Cell B: 0.6 : 1. The smaller cell has the larger ratio.

Example 2: Agar Cubes and Dye

Agar cubes of different sizes were soaked in dye until it reached 0.5 cm in from every face (sample data). Write a claim, evidence, and reasoning about how cube size affects the part of the cube the dye reaches.

Side of cube (cm)	Percent of volume the dye reached
1	100
2	87.5
3	70.4
4	57.8
6	42.1

Show the solution

1. Claim: the larger the cube, the smaller the percent of its volume the dye reaches.
2. Evidence: the dye reached 100% of the 1 cm cube, 87.5% of the 2 cm cube, 70.4% of the 3 cm cube, 57.8% of the 4 cm cube, and only 42.1% of the 6 cm cube.
3. Reasoning: a larger cube has a smaller SA : V ratio, so less of its volume is near the surface. In a cell, the same pattern would leave parts of a large cell short of supplies.
4. Answer: Larger cubes were reached less completely, because their SA : V ratio is smaller.

Example 3: Use the Diffusion Model

Use the model to compare the time for a small molecule to diffuse across a cell 10 µm wide and across a cell 1,000 µm wide. Time = distance² / (2 x D), with D = 1,000 µm²/s (a typical value for a small molecule in water).

Show the solution

1. 10 µm: t = (10 µm)² / (2 x 1,000 µm²/s) = 100 / 2,000 s = 0.05 s.
2. 1,000 µm: t = (1,000 µm)² / (2 x 1,000 µm²/s) = 1,000,000 / 2,000 s = 500 s, about 8 minutes.
3. Compare: the distance is 100 times longer, and the time is 10,000 times longer (500 / 0.05).
4. Answer: 0.05 s for the 10 µm cell and 500 s (about 8 minutes) for the 1,000 µm cell.

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Practice

Calculate surface area, volume, and SA : V ratios, read data from an agar cube experiment, work out how doubling the side changes the ratio, and explain how microvilli help. Try it on your own first, then check your work with the answer key.

• Surface Area to Volume Ratio Practice Worksheet (PDF)
• Surface Area to Volume Ratio Practice Answer Key (PDF)

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Lesson Slides

• 6.8 Surface Area to Volume Ratio Slides (PDF)

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Big Ideas

• For a cube, SA = 6s², V = s³, and SA : V = 6 / s
• As size increases, volume grows faster than surface area, so SA : V falls
• Diffusion time grows with the square of the distance
• Small cells keep all of their volume close to the surface
• Microvilli and other folds increase surface area

Back to the essential question: as a cell gets bigger, its volume increases faster than its surface area, so the SA : V ratio falls and materials have farther to travel. That is why most cells are small, and why cells that need a lot of surface area have folds and projections. Next, you’ll see how specialized cells work together in organ systems.

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