Surface Area to Volume Ratio

Surface Area to Volume Ratio

Although it may seem to be just a mathematical activity, the surface area to volume ratio actually has a real world significance. In this lesson, you'll learn how you can calculate this ratio along with how useful this ratio is useful in various aspects of science.

Surface Area

All shapes, including our bodies, have a surface area. Surface area is the area of the object that is exposed on the outside. So, for your body, it's how much skin you have. For a cube, it's the total area of all six sides of the cube. In math, geometric shapes have specific formulas you can use to calculate their surface areas.

A cube has a surface area of 6s2 and a sphere has a surface area of 4 * pi * r2.

Volume

These same three-dimensional geometric shapes also have formulas for volume. The volume is the space inside the shape. Only three-dimensional shapes have volume. Two-dimensional shapes, such as a line or a square, don't have volume. If it helps, you can think of volume as how much liquid would fit inside a shape.

The formula for the volume of a cube is s3.

For the sphere, the volume is 4/3 * pi * r3.

These formulas make it very easy to find the surface area and volume of these shapes. For other shapes, more complicated math is such as integral calculus may be needed. For this lesson, though, you won't need to know calculus, just the formulas that are available for common shapes.

Surface Area to Volume Ratio

When you divide your surface area by the volume, you get a special ratio called the surface area to volume ratio. This ratio can be noted as SA:V. This ratio is not just a mathematical exercise, but a useful real world ratio. In chemistry, this ratio is used to determine how quickly a chemical will dissolve. The higher the ratio, the quicker the chemical will dissolve. In biology, this ratio tells you how much exposure a cell or organism has to the environment. This is important in cells or single cell organisms in regard to the transport of materials across cell walls.

To find this ratio for common geometric shapes, you can divide the formula for surface area by the formula for volume. Or, if you are given the two numbers directly, you can simply divide the surface area number by the volume number.

For example, to calculate the surface to volume ratio for the cube using the formulas, you divide the formula for surface area by the surface for volume, then you simplify as much as you can. This is what you get.

To use this ratio, you plug in the value for your variable and then calculate your ratio. So, if your cube has a side length of 1 centimeter with a volume of 1 cubic centimeter, then your surface area to volume ratio is 6 / 1 = 6. The only thing you needed to do is to plug in your value for the length of your side.

Example

Let's try calculating the surface area to volume ratio for the sphere.

To calculate this ratio for the sphere, you do the same as you did for the cube. You take the formula for surface area and you divide it by the formula for volume. This is what you get.

You use this ratio the same as you would for the cube. If you have a radius of 1 centimeter for a volume of 4.187 cubic centimeters, then your surface area to volume ratio is 3 / 1 = 3. All you have to do is plug in your value for the radius.

Lesson Summary

Surface area is how much area of the object is exposed to the outside. The volume is how much space is inside the shape. The surface area to volume ratio tells you how much surface area there is per unit of volume. This ratio can be noted as SA:V. To find this ratio, you divide the formula for surface area by the formula for volume and then you simplify. If you are given the numbers, then you simply divide the surface area number by the volume number.