What this calculator does
The surface area to volume ratio compares how much outer surface a shape has against how much space it encloses. As a shape gets bigger while keeping the same proportions, its volume grows faster than its surface area, so the ratio falls. A small object is mostly surface; a large one is mostly interior.
This matters well beyond geometry class. A cell needs enough surface area to exchange nutrients and waste relative to the volume of cytoplasm it has to service, which is one reason biological cells stay small. An animal loses heat through its surface but generates it throughout its volume, so a small animal cools faster than a large one of the same shape. Enter the shape and its size and the ratio, surface area and volume all come out together.
The formula
Surface area (SA) and volume (V) each have their own formula per shape: a cube of side s has SA = 6s² and V = s³; a sphere of radius r has SA = 4πr² and V = (4/3)πr³; a cylinder of radius r and height h has SA = 2πr(r + h) and V = πr²h. The ratio is simply SA divided by V, which for a single dimension collapses to a clean expression: 6/s for a cube, 3/r for a sphere.
| Term | Meaning |
|---|---|
| SA:V ratio | Surface area divided by volume, in units of 1 ÷ length. |
| s | The side length of a cube. |
| r | The radius of a sphere or cylinder. |
| h | The height of a cylinder (ignored for a cube or sphere). |
The inputs explained
| Field | What to enter |
|---|---|
| Shape | Choose which solid to calculate: cube, sphere or cylinder. |
| Side (cube) or radius (sphere/cylinder) | The side length for a cube, or the radius for a sphere and cylinder. |
| Height (cylinder only) | The height, used only when the shape is a cylinder. |
When to use it
Why small cells work and huge ones do not
A cell relies on surface area to move nutrients and waste across its membrane fast enough to serve the volume inside. As a cell grows, volume outpaces surface area, and past a certain size diffusion can no longer keep up, which is why cells divide rather than simply growing indefinitely.
Heat loss in small versus large animals
A small animal has a high surface area to volume ratio, so it loses body heat quickly relative to the heat its mass can generate, which is why small mammals tend to have faster metabolisms and eat proportionally more than large ones.
Engineering and materials
Crushing a solid into smaller pieces or grains increases its total surface area to volume ratio dramatically, which is why powdered or granular materials react, dissolve or combust faster than a single solid block of the same total volume.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the surface area to volume ratio change as a cube gets bigger?
The same shape, cube, at increasing side lengths.
| Side length | Surface-to-volume ratio | Surface area |
|---|---|---|
| 1 units | 6.000 per unit length | 6.000 sq. units |
| 2 units | 3.000 per unit length | 24.000 sq. units |
| 4 units | 1.500 per unit length | 96.000 sq. units |
| 8 units | 0.7500 per unit length | 384.000 sq. units |
| 16 units | 0.3750 per unit length | 1,536.00 sq. units |
| 32 units | 0.1875 per unit length | 6,144.00 sq. units |
How do a sphere and a cylinder of the same radius compare?
A sphere of radius 3 alongside a cylinder of the same radius at a few different heights.
| Cylinder height | Surface-to-volume ratio | Volume |
|---|---|---|
| 1 units | 2.667 per unit length | 28.274 cubic units |
| 3 units | 1.333 per unit length | 84.823 cubic units |
| 6 units | 1.000 per unit length | 169.646 cubic units |
| 12 units | 0.8333 per unit length | 339.292 cubic units |
| 24 units | 0.7500 per unit length | 678.584 cubic units |
Questions
What is surface area to volume ratio?
It is the surface area of a shape divided by its volume, a measure of how much outer boundary a shape has relative to the space it fills. A high ratio means a lot of surface for the volume enclosed; a low ratio means comparatively little.
How do you calculate surface area to volume ratio?
Work out the surface area and volume of the shape separately using its own formula, then divide the surface area by the volume. For a cube of side s this reduces to 6 ÷ s, and for a sphere of radius r it reduces to 3 ÷ r.
Why does the ratio fall as a shape gets bigger?
Surface area scales with the square of a linear dimension while volume scales with the cube of it, so volume grows faster than surface area as a shape is scaled up. The ratio, being surface area over volume, therefore falls as size increases, for any fixed shape.
Which shape has the lowest surface area to volume ratio?
For a fixed volume, a sphere has the lowest possible surface area of any shape, which is why spheres appear so often in nature where minimising surface relative to volume is an advantage, such as in bubbles, droplets and some cell types.
To scale a known area and volume by a linear factor instead of computing them from scratch, see the area and volume scaling calculator. For the volume of a sphere on its own, use the sphere volume calculator.