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## 'Pick's Theorem' printed from http://nrich.maths.org/

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 shape $(6,4)$:

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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.

Do you agree?

Draw more figures; tabulate the information about their perimeter points ($p$), interior points ($i$) and their areas ($A$).

Can you find a relationship between all these three variables ($p$, $i$ and $A$)?

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