I imagine pages 26-29 of your lab books is giving some of you a bit of a head-ache.....let me try to help you out.....(this is a long post, I recommend a mug of hot chocolate!)

This page is all about how shape, volume and mass can be related to each other.

Lets start with shape and volume:

You can measure the shape of something by measuring lengths (L) i.e. height, width, diameter etc - all of these measurements will help define the shape of an object/organism. 

What happens if the shape of an object stays the same, but the size (i.e. volume) gets bigger? Well, all of the lengths will get bigger by the same amount i.e. they might all double. This means that the measurements will all be proportional to each other:

image modified from: http://simple.wikipedia.org/wiki/Cuboid

So in the above example, the lengths have all doubled. Because the shape has stayed the same even though the volume has increased, the lengths have kept the same proportions relative to each other i.e. the length of the small cubiod (8 cm) is double its height (4 cm), and the length of the large cuboid (16 cm) is also double its height (8 cm). 

So, how are the lengths you can measure related to the volume of an object? Well, the volume of something is proportional to any of it's dimensions cubed (i.e. L³). This is quite obvious when you consider a simple shape like a cube:

image taken from: http://www.ableweb.org/volumes/vol-20/1-colton/scalingtutorial/geometric.html

For each of these cubes the volume can be calculated by multiplying the length of one side 3 times (i.e. L x L x L or L³). So, V = L³.

That is a simple example....what about for a sphere? Well, we have all learnt at school the equation for calculating the volume of a sphere:

In this equation, the radius of the sphere (r) is the length that is being cubed (r³). We also have the 4/3π bit. Now this is called the 'shape-coeffiecient', which in your lab-books is given the letter k. For a sphere, k = 4/3π. What k is will be decided by the shape of the object and the length you have chosen.

Hang on, you might be wondering what was going on with the cube example, where we said that the Volume = Length³ - where is k in this example!? Well, for a cube, if the length you are using is the length of one of the sides of the cube, then k = 1, so we don't need to put it in to the equation. 

So, what I am trying to demonstrate, is that the Volume of ANY object can be found using the equation

And what value k has depends on two things: the shape of the object and the length you are using for L.

So far we have only been looking at the relationship between shape and volume. Now we need to think about the relationship between mass and volume

The equation for mass is:

Mass (M) = Density (ρ) x Volume (V)

or

M = ρV

Now we can write this equation a little bit differently, if instead of V we insert kL³. (remember V = kL³):

So how would we actually use all of this in Biology?

Well, imagine we are dealing with an animal that when it grows keeps the same dimensions and also the same density, like the salamander:

In this case, where shape and density stay the same even though the animal is getting bigger, we can say that k, the shape coefficient, and ρ, the density, will always have the same value when we want to calculate the Mass of any of these lizards using the equation:

i.e. ρ and k will be constant.

This means that if you can measure one dimension of all of the above salamanders, and if you know ρ and k, you can calculate the mass of each of them.

Ok ok, so how do you know what ρ and k actually are??

Well, you could get lots of Salamanders and measure one dimension (eg their lengths) and their masses. You could then plot this on a graph - Mass on the y axis and length³ on the x axis:

If you did this you would get a straight line going through zero (no length means no mass right!)

The gradient of this line will be your value of ρk.

WHY is the gradient ρk?? Remember the equation for a straight line:

Ok, so you don't have a value for ρ and k individually, but you do have a value for what ρ times k would equal. 

Phew, that was a lot! I really hope this helps!!