Area and Volume

Let’s say you have two similar prisms, A and B. The surface area of A is 161 square centimeters, and its volume is 1331 cubic centimeters. The surface area of B is 250 square centimeters. Can you find the volume of prism B?

It seems like you need more information. You don’t even know how many sides the prism bases have! It turns out there’s a neat principle involving areas in general and volumes that we can use here, and we don’t need to know any specifics about the prisms beyond their similarity. The short answer is, volume is in sesquialterate ratio with surface area. The long answer is the rest of this post!

First, let’s get clear what prisms are, and what it means to have similar prisms.

A prism is a spatial object with two identical, flat faces (the bases) that, when cut anywhere along its length parallel to those faces, has identical cross sections throughout. The general image of a prism has two triangles for bases, and three rectangular faces connecting the triangles. Prisms like this made of glass are generally used to split white light into its spectrum of colors.

Here, light is passing through the identical faces, and the prism sits on one of its two triangular bases. In general, prisms can have any shape base, and any length of face. The simplest prism is actually the cylinder, about which we shall see more below.

Similar prisms are defined just like any other similar shapes. In a picture, they look identical unless they’re either right next to each other, or both compared with a known measure. Visually, the larger similar figure just looks like it’s just the smaller figure brought closer to you. In other words, all lengths found on one prism are in constant proportion with all comparable lengths on a similar prism.

Here, we have two similar rectangular prisms. The larger one has lengths $X, Y,$ and $Z$, while the smaller one has lengths $\alpha, \beta,$ and $\gamma$. Since they’re similar, each pair of comparable sides are proportional with each other. If our constant of proportionality is $\kappa$, then we have

$X = \kappa \alpha$
$Y = \kappa \beta$
$Z = \kappa \gamma$.

The surface areas of the two prisms are $2XY + 2YZ + 2ZX$ and $2\alpha\beta + 2\beta\gamma + 2\gamma\alpha$. The two volumes are $XYZ$ and $\alpha\beta\gamma$. Since the two prisms are similar, we can relate the surface areas with

$2XY + 2YZ + 2ZX = 2 \kappa\alpha \kappa\beta + 2 \kappa\beta \kappa\gamma + 2 \kappa\gamma \kappa\alpha = \kappa^2 (2\alpha\beta + 2\beta\gamma + 2\gamma\alpha)$

In other words, the surface area of the larger prism is equal to $\kappa^2$ times the surface area of the smaller one. There’s a similar relationship between the volumes

$XYZ = \kappa\alpha \kappa\beta \kappa\gamma = \kappa^3 \alpha\beta\gamma$,

or, the volume of the larger prism is $\kappa^3$ times the volume of the smaller one. More concisely,

$\frac{SA_{large}}{SA_{small}} = \kappa^2$
$\frac{V_{large}}{V_{small}} = \kappa^3$

The constant of proportionality, $\kappa$, can be thought of as a magnification or scaling factor. This means that, for example, if we double the size of a prism, the surface area gets $2^2=4$ times larger, while the volume gets $2^3=8$ times larger. In other words, surface area grows as a square, while volume grows as a cube.

We just did this for two similar rectangular prisms. Does this hold for all prisms? Remember the prism principle – two sides must be parallel and identical, the other sides just connect these two together. The main difficulty in determining surface area or volume in a prism lies in the shape of the two parallel bases. When you double an oddly shaped base’s sides, the area of the base does not double, but rather quadruple

Meno ran into this when he was trying to double the square for Socrates. Doubling area doesn’t follow from doubling side length. When you increase the length of a polygon’s side, the volume increases by the square. This is why a larger similar prism will always have a total surface area that is larger than that of a smaller prism by the square of the scaling factor. With volumes (I won’t try to draw this one), as you increase side length, you increase the volume by the cube. Try it with blocks.

Therefore, the shape of the prisms doesn’t matter. The ratio of volumes and the ratio of surface areas only depends on the scaling factor. In fact, we can use this to find a direct relationship between the surface areas and volumes:

$\frac{SA_{large}}{SA_{small}} = \kappa^2 \Rightarrow (\frac{SA_{large}}{SA_{small}})^{\frac{1}{2}} = \kappa$
$\frac{V_{large}}{V_{small}} = \kappa^3 \Rightarrow (\frac{V_{large}}{V_{small}})^{\frac{1}{3}} = \kappa$
or,
$(\frac{SA_{large}}{SA_{small}})^{\frac{1}{2}} = \kappa = (\frac{V_{large}}{V_{small}})^{\frac{1}{3}}$

This relationship, where the square of one side is equal to the cube of the other, is called a “sesquialterate ratio.” Some may remember this type of relationship from Kepler’s third law.

In any case, now we are able to solve our problem. The smaller prism has surface area 161 square centimeters and volume 1331 cubic centimeters, while the larger prism has surface area 250 square centimeters. To find the volume of the larger prism, we just rearrange our sesquialterate ratio

$(\frac{SA_{large}}{SA_{small}})^{\frac{1}{2}} = (\frac{V_{large}}{V_{small}})^{\frac{1}{3}}$
$(\frac{SA_{large}}{SA_{small}})^{\frac{3}{2}} = \frac{V_{large}}{V_{small}}$
$V_{small} (\frac{SA_{large}}{SA_{small}})^{\frac{3}{2}} = V_{large}$

Putting in the numbers,

$V_{large} = (1331 cm^3) (\frac{250 cm^2}{161 cm^2})^{\frac{3}{2}} = 2575.4 cm^3$

Pretty easy, right? Maybe not too easy. Still, the fact that volume and surface area know to always follow their sesquialterate relationship is another testament to our incredibly constructed universe!