Areas on a Grid

Take a look at the shapes on the interactivity below, and then draw some of your own.
You might want to print off some dotty paper.

Can you work out the area of the shapes?

I wonder if there is a relationship between the number of dots and the area...

Take a look at the video below showing some shapes and their areas:

Can you find other polygons with one dot inside and an area of 2 square units?

Can you find other polygons with two dots inside and an area of $3\frac{1}{2}$ square units?

Can you find a relationship between the number of dots on the edge of a polygon, the number of dots inside, and the area?

Picks Theorem

You can also use Picks Theorem:

If the grid is square, count the number of grid intersections lying on the perimeter ($b$) and the number of grid intersections on the inside ($c$). Pick's Theorem says that the area is $\frac{b}{2}+c-1$ where the unit is the size of the unit square of the grid.

The technique was used by land agents to estimate the size of irregular fields in the later part of the 19th century.

Picks Theorem

Very well done! Can you find any shapes for which Pick's Theorem does not work? What assumptions do we make about the shapes in order for Pick's Theorem to work?