# Find the surface area to volume ratio

Finds the surface area, the volume and the surface area to volume ratio (SA:V) of a cube, a sphere, a cylinder or a rectangular box, with the steps.

- Page: https://www.acalculator.org/math/surface-area-to-volume-ratio-calculator
- JSON spec: https://www.acalculator.org/math/surface-area-to-volume-ratio-calculator.json
- Version: feda43913118

## Default answer

Example with the default inputs (Shape Cube, Side (a) 2, Length unit cm): The surface area to volume ratio is 3 : 1 (3 per cm).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| shape | Shape | The shape: a cube, a sphere, a cylinder or a rectangular box. |
| a | Side (a) | The length of one edge of the cube. |
| r | Radius (r) | The radius: half the diameter. |
| l | Length (l) | The length of the box. |
| w | Width (w) | The width of the box. |
| h | Height (h) | The height of the cylinder or the box. |
| u | Length unit | The unit of the lengths you typed; the area is in its square and the volume in its cube. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| result | Surface area to volume ratio | SA:V as a ratio to 1, and per unit of length. |
| ratio | Surface area to volume ratio (per unit of length) | The surface area divided by the volume. |
| area | Surface area (unit²) | The total area of the outside, in the typed unit squared. |
| volume | Volume (unit³) | The space inside, in the typed unit cubed. |
| steps | Steps | The surface area, the volume and the ratio worked out. |

## Method

Cube: SA = 6a², V = a³, SA:V = 6 ÷ a. Sphere: SA = 4πr², V = 4/3 πr³, SA:V = 3 ÷ r. Cylinder: SA = 2πr² + 2πrh, V = πr²h, SA:V = 2(r + h) ÷ (rh). Box: SA = 2(lw + lh + wh), V = lwh.

## Assumptions

- All lengths are in the same unit; the ratio is per that unit of length, so it changes when the unit changes.
- Typed lengths are read as the exact decimals you typed.

## Worked examples

1. shape = cube, a = 1, u = mm gives area = 6, volume = 1, ratio = 6, result = 6 : 1 (6 per mm). Source: OpenStax, Biology 2e, §4.2 Prokaryotic Cells (Cell Size: a sphere’s surface area is 4πr² and its volume 4πr³/3; Figure 4.7: cells of 1 mm³ and 6 mm² and of 8 mm³ and 24 mm² have surface area to volume ratios of 6 to 1 and 3 to 1), https://openstax.org/books/biology-2e/pages/4-2-prokaryotic-cells (retrieved 2026-10-19); OpenStax, Prealgebra 2e, §9.6 Solve Geometry Applications: Volume and Surface Area (rectangular solid V = LWH, S = 2LH + 2LW + 2WH; cube V = s³, S = 6s²; sphere V = 4/3 πr³, S = 4πr²; cylinder V = πr²h, S = 2πr² + 2πrh), https://openstax.org/books/prealgebra-2e/pages/9-6-solve-geometry-applications-volume-and-surface-area (retrieved 2026-10-19).
2. shape = cube, a = 2, u = mm gives area = 24, volume = 8, ratio = 3, steps = Surface area = 6a² = 6 × 2² = 24 mm²; Volume = a³ = 2³ = 8 mm³; Ratio = 6 ÷ a = 6 ÷ 2 = 3 per mm. Source: OpenStax, Biology 2e, §4.2 Prokaryotic Cells (Cell Size: a sphere’s surface area is 4πr² and its volume 4πr³/3; Figure 4.7: cells of 1 mm³ and 6 mm² and of 8 mm³ and 24 mm² have surface area to volume ratios of 6 to 1 and 3 to 1), https://openstax.org/books/biology-2e/pages/4-2-prokaryotic-cells (retrieved 2026-10-19); OpenStax, Prealgebra 2e, §9.6 Solve Geometry Applications: Volume and Surface Area (rectangular solid V = LWH, S = 2LH + 2LW + 2WH; cube V = s³, S = 6s²; sphere V = 4/3 πr³, S = 4πr²; cylinder V = πr²h, S = 2πr² + 2πrh), https://openstax.org/books/prealgebra-2e/pages/9-6-solve-geometry-applications-volume-and-surface-area (retrieved 2026-10-19).
3. shape = cube, a = 2.5, u = in gives area = 37.5, volume = 15.625, ratio = 2.4. Source: OpenStax, Prealgebra 2e, §9.6 Solve Geometry Applications: Volume and Surface Area (rectangular solid V = LWH, S = 2LH + 2LW + 2WH; cube V = s³, S = 6s²; sphere V = 4/3 πr³, S = 4πr²; cylinder V = πr²h, S = 2πr² + 2πrh), https://openstax.org/books/prealgebra-2e/pages/9-6-solve-geometry-applications-volume-and-surface-area (retrieved 2026-10-19).
4. shape = sphere, r = 3, u = um gives area = 113.097, volume = 113.097, ratio = 1, result = 1 : 1 (1 per µm). Source: OpenStax, Biology 2e, §4.2 Prokaryotic Cells (Cell Size: a sphere’s surface area is 4πr² and its volume 4πr³/3; Figure 4.7: cells of 1 mm³ and 6 mm² and of 8 mm³ and 24 mm² have surface area to volume ratios of 6 to 1 and 3 to 1), https://openstax.org/books/biology-2e/pages/4-2-prokaryotic-cells (retrieved 2026-10-19).
5. shape = cylinder, r = 1, h = 2, u = cm gives area = 18.8496, volume = 6.28319, ratio = 3. Source: OpenStax, Prealgebra 2e, §9.6 Solve Geometry Applications: Volume and Surface Area (rectangular solid V = LWH, S = 2LH + 2LW + 2WH; cube V = s³, S = 6s²; sphere V = 4/3 πr³, S = 4πr²; cylinder V = πr²h, S = 2πr² + 2πrh), https://openstax.org/books/prealgebra-2e/pages/9-6-solve-geometry-applications-volume-and-surface-area (retrieved 2026-10-19).
6. shape = box, l = 3, w = 2, h = 1, u = in gives area = 22, volume = 6, ratio = 3.66667, result = 3.66667 : 1 (3.66667 per in). Source: OpenStax, Prealgebra 2e, §9.6 Solve Geometry Applications: Volume and Surface Area (rectangular solid V = LWH, S = 2LH + 2LW + 2WH; cube V = s³, S = 6s²; sphere V = 4/3 πr³, S = 4πr²; cylinder V = πr²h, S = 2πr² + 2πrh), https://openstax.org/books/prealgebra-2e/pages/9-6-solve-geometry-applications-volume-and-surface-area (retrieved 2026-10-19).

## FAQ

### How do you calculate the surface area to volume ratio?

Work out the surface area and the volume, then divide: SA:V = surface area ÷ volume. A cube with 2 mm sides has 6 × 2² = 24 mm² and 2³ = 8 mm³, so SA:V = 24 ÷ 8 = 3, written 3 : 1.

### What is the surface area to volume ratio of a sphere?

SA:V = 4πr² ÷ (4/3 πr³) = 3 ÷ r: π cancels. A cell of radius 3 µm has SA:V = 1 per µm; at radius 1 µm it is 3 per µm.

### Why does the ratio fall as a cell gets bigger?

Area grows with the square of the size and volume with the cube (OpenStax Biology 2e §4.2). Double a cube’s side and its area grows 4 times but its volume 8 times, so SA:V halves: 6 : 1 at 1 mm, 3 : 1 at 2 mm.

### Why does a high surface area to volume ratio matter for cells?

A cell takes in food and gets rid of waste through its surface, but its needs grow with its volume. A small cell has more surface for each unit of volume, which is one reason cells stay small or divide.

### Does the unit change the ratio?

Yes. SA:V has the unit 1 ÷ length: a 1 cm cube has 6 per cm, which is 0.6 per mm. Keep every length in one unit and read the ratio per that unit.

### What is the ratio for a cylinder?

SA = 2πr² + 2πrh and V = πr²h, so SA:V = 2(r + h) ÷ (rh). A cylinder of radius 1 cm and height 2 cm has 2 × 3 ÷ 2 = 3 per cm.

## Sources

- OpenStax, Biology 2e, §4.2 Prokaryotic Cells, Cell Size: a sphere’s surface area is 4πr² and its volume 4πr³/3; Figure 4.7: cells of 1 mm³ with 6 mm² and of 8 mm³ with 24 mm² have surface area to volume ratios of 6 to 1 and 3 to 1, retrieved 2026-10-19. https://openstax.org/books/biology-2e/pages/4-2-prokaryotic-cells
- OpenStax, Prealgebra 2e, §9.6 Solve Geometry Applications: Volume and Surface Area: rectangular solid V = LWH, S = 2LH + 2LW + 2WH; cube V = s³, S = 6s²; sphere V = 4/3 πr³, S = 4πr²; cylinder V = πr²h, S = 2πr² + 2πrh; Example 9.49 (a cube of 2.5 in: V = 15.625 in³, S = 37.5 in²), retrieved 2026-10-19. https://openstax.org/books/prealgebra-2e/pages/9-6-solve-geometry-applications-volume-and-surface-area