Pick's Theorem

Stage: 3 Challenge Level: Challenge Level:1

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 $(6,4)$:

This text is usually replaced by the Flash movie.


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 $A$)?

A poster of this problem can be found here.


search engine page

Published October 2003,March 2009,April 2009.