The shoelace formula (Gauss's area formula) simply allows you to calculate the area of any $n$ sided polygon given the coordinates of its points. It is defined as:

$$A=\frac12\Big|\sum_{i=1}^nx_i(y_{i+1}-y_{i-1})\Big|$$

Where $A$ is the area of the polygon, $n$ is the number of sides of the polygon and $(x_i,y_i)$ are the vertices of the polygon.

Whilst this formula doesn't do anything that can't be achieved by other methods, it does make solving problems a lot easier in many cases.

## Explanation

Whilst it isn't necessary to be able to prove something to make it useful, I generally find that it is a good plan to have a vague idea about what is going on.

What is actually happening is that for every line $AB$, the signed area of the triangle $ABO$ is being calculated. This means that when two triangles overlap, the overlapping area is cancelled out which leaves only the area of the polygon.

## Example

A triangle has points $(a,2a),(2a,4a),(4a,0)$ and area $64$ what is the value of $a$?

Normally this would be fairly difficult and may involve drawing a diagram and boxing off the triangle or calculating the side lengths and using Heron's formula however thanks to the shoelace formula it becomes trivial:

$$\frac12|a(4a-0)+2a(0-2a)+4a(2a-4a)|=64$$

$$\frac12|4a^2-4a ^2-8a ^2|=64$$

$$4a^2=64$$

$$a=\pm4$$