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# Pick's Theorem

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Age 14 to 16

Challenge Level

*Pick's Theorem printable worksheet
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?
Can you find the area of any polygon once you know the number of perimeter points and interior points?**

Click here for a poster of this problem.