# Areas on a grid

Take a look at the video showing areas of different shapes on dotty grids...

*This resource is part of "Dotty Grids - Exploring Area"*

I wonder what we can find out about the areas of shapes drawn on dotty grids...

This video might give you some ideas.

Play with the dotted grid below, or print out some dotty paper.

Kathryn from Garden International School noticed two relationships:

Nadia from Melbourn Village College, Yun from Garden Internation School, Simeran and Aaron from Woodfield Junior School, and Kahlia from Merici College all discovered that:

When

$A=$ the area of the shape,

$p=$ the number of dots on the perimeter and

$i=$ the number of dots inside the shape,

the area is equal to half of $p$, added to $i$, minus $1$

$$A=p/2+iâˆ’1$$

Matthew from St Anthony's Catholic College, Australia and Yash from the UK both knew a theroem which could help them.

Picks theorem states that the area of a polygon in units squared can be

calculated by the formula:

$$A = i + \frac{b}{2} -1$$

Where $A$ is the area of the polygon, $i$ is the interior points that lie

inside the polygon and $b$ is the boundary points of the polygon.

Pick's theorem is quite difficult to prove, but here is a resource which could help you work through it.

Well done to everyone who sent in a solution. Keep exploring!

As the number of dots on the shape's perimeter increases by one, the area increases by half.

As the number of internal dots increases by one, the area also increases by one.Nadia from Melbourn Village College, Yun from Garden Internation School, Simeran and Aaron from Woodfield Junior School, and Kahlia from Merici College all discovered that:

When

$A=$ the area of the shape,

$p=$ the number of dots on the perimeter and

$i=$ the number of dots inside the shape,

the area is equal to half of $p$, added to $i$, minus $1$

$$A=p/2+iâˆ’1$$

Matthew from St Anthony's Catholic College, Australia and Yash from the UK both knew a theroem which could help them.

Picks theorem states that the area of a polygon in units squared can be

calculated by the formula:

$$A = i + \frac{b}{2} -1$$

Where $A$ is the area of the polygon, $i$ is the interior points that lie

inside the polygon and $b$ is the boundary points of the polygon.

Pick's theorem is quite difficult to prove, but here is a resource which could help you work through it.

Well done to everyone who sent in a solution. Keep exploring!

For ideas on how this problem and others from the Dotty Grids Collections can be used in the classroom, you may be interested to read this article.

A printable version of this problem is available as a Word or Pdf file.