Areas on a Grid

Kathryn from Garden International School noticed two relationships:

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 $1$


$$A=p/2+i−1$$


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!