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# Pick's Theorem

##### Age 14 to 16 Challenge Level:

To work on this problem you may want to print out some dotty paper

When the dots on square dotty paper are joined by straight lines the enclosed figures have dots on their perimeter ($p$) and often internal ($i$) ones as well.

Figures can be described in this way: $(p, i)$.
For example, the red square has a $(p,i)$ of $(4,0)$, the grey triangle $(3,1)$, the green triangle $(5,0)$ and the blue hexagon $(6,4)$: Each figure you produce will always enclose an area ($A$) of the square dotty paper.

The examples in the diagram have areas of $1$, $1 {1 \over 2}$, and $6$ sq units.

Check that you agree.

Draw more figures and keep a record of their perimeter points ($p$), interior points ($i$) and areas ($A$).

Can you find a relationship between these three variables?