Areas and Volumes - A Lesson in Proportionality and the Value of Ratios

In astronomy and planetary science we often need to consider areas and volumes. We might be interested in the light hitting the surface of a planet or the distribution of craters over the surface - how many craters per unit area, for example. Then, there are processes that depend on the volume - the amount of heat generated by radioactivity inside, for example.

First, let's start with something easy - squares and cubes:

Consider a square that is some distance "D" on a side - the area is D x D= D2 (with the appropriate units - meters2 or km2 or feet2 (if you cannot handle metric))

Now, double the size of the square so that each side is 2D - the area is then 2D x 2D = (2D)2 = 4 x D2 (with same appropriate units).

The ratio of sizes of the squares is (Size Big)/(Size Small) = (2D)/(D) = 2.

The ratio of the areas of the squares is (Area Big)/(Area Small) = (2D)2 / D2 = 22 = 4.

Next , let's look at volume. Consider a volume of a cube that is D on a side. The volume of the cube is D x D x D = D3 (with the appropriate units - meters3 or km3 or feet3 (if you cannot handle metric))

Again, consider a cube that is twice the size - 2D on a side. The volume of the cube is then (2D)3 = 23 D3 = 8 D3.

Taking ratios we can summarize as follow (labelling the square or cube that is of size of D as "small" and that the one that is twice the size, 2D, as "big"):

(Size Big)/(Size Small) = (2D)/ D = 2

(Area Big)/(Area Small) = (2D)2/ D2 = 22 = 2 x 2 = 4 = [(Size Big)/(Size Small)]2

(Volume Big)/ (Volume Small) = (2D)3/ D3 = 23 = 2 x 2 x 2 = 8 = [(Size Big)/(Size Small)]3

OK - but planets and stars are spheres not cubes. But let's begin with a circle of radius R. The area =p R2.

If we double the size of the circle - double the radius (and/or double the diameter) - Area =p (2R)2 = =p 22 R2 = 4 R2

Moving from 2 dimensions to 3 dimensions, let's thing about spheres. Remember the formula for the area of a sphere of radius R? It is 4p R2

Let's go for a sphere twice the size - radius = 2D. The area will then be 4p (2R)2 = 4p 22R2 = 16 pR2

Volume? The volume of the smaller sphere is 4/3 pR3

Double the radius of the sphere and we have a volume of 4/3 p(2R)3= 4/3 p23R3= 4/3 x 8 pR3= 32/3 pR3

Taking ratios again, we get

(Area Big Circle)/ (Area Small Circle) = (p (2R)2) / ( p R2) = 22= 4

(Area Big Sphere)/ (Area Small Sphere) = (4p (2R)2) / ( 4p R2) = 22= 4

(Volume Big Sphere)/ (Volume Small Sphere) = (4/3 p (2R)3) / ( 4/3 p R3) = 23= 8

OK - so, you have mastered squares, cubes, circles and spheres - let's move on to frogs. What's the formula for the area of a frog?

Area of a frog = "something" x H2

What if the frog is twice as big? Area of big frog = "something" x (2H)2= "something" x 22H2 = "something" x 4 H2

And, the volume of an frog?

Volume of a frog = "something else" x H3

Volume of the big frog = "something else" x (2H)3= "something else" x 23H3= "something else" x 8 H3

But, while we might have no idea about the "something" or the "something else", look what happens when we take ratios:

(Area Big Frog)/ (Area Small Frog) = (something x (2H)2) / (something x H2) = 22= 4

(Volume Big Frog)/ (Volume Small Frog) = (something else (2H)3) / (something else H3) = 23= 8

So, What do we conclude from all this? That if you are not interested in the actual formula for an area of volume of an object, whatever its shape, but interested in what happens when you consider an object of the same shape but some factor bigger or smaller, then all you need to think about is the ratio.

Furthermore, if you look at the above, you will see a pattern. You will see that we can generalize. There are trends. You will notice that the ratio of areas is always (Area Big)/ (Area Small) = [(Size Big)]/(Size Small)]2. And that

(Volume Big)/(Volume Small) = [(Size Big)]/(Size Small)]3.

In the above cases we always doubled the size of the object (to keep it simple) - but the math would be the same if we multiplied by 3 or 8 or 42.456788. The area would then increase by a factor of 32 or 82 or 42.4567882 and the volume by a factor of 33 or 83 or 42.4567883.

We can say

Area is proportional to Size2

Volume is proportional to Size3

This works for squares, this works spheres, this works for frogs, elephants, you name it. Furthermore - it means that if you are just looking at the same shape and scaling bigger or smller, you never need to plug p into your calculator. In fact, you can probably do all the math in your head. It also means that you are less likely to hit the wrong key on your calculator.

For example, the marbles problem. Jupiter is 10 times the size of Earth. So, if we model Jupiter with a volley ball and Earth with a marble, how many marbles fit into the Jupiter-volley ball? The ratio of volumes is just

(Volume of Jupiter)/(Volume of Earth) = [(radius of Jupiter)/(radius of earth)] 3 = (10)3 = 1000

BUT.... IN REALITY..... there will be space between the marbles so that we will probably not be able to get as much as 1000 marbles into the Jupiter ball - there will be gaps. How much? You guess! 10%? 20%? 30%? 50%....? what do you think?