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Pick's Theorem

Age 11 to 14 Challenge Level:

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?

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