Area enclosed by a string on a lattice

First find the area of the square with green boundary which is (13 * 12=156).

Now subtract the area of the 3 triangle and one trapezium which are labelled as 1,2,3 and 4 respectively . The areas are:

(1) 8 * 2/2=8

(2) 10 * 5/2=25

(3) (5+1) * 8/2=24

(4) 4 * 1/2=2

The area enclosed by the string is 156- 8 - 25 -24 - 2=97

solving subproblems

Use Pick's theorem for this

(This theorem was discovered by Georg Alexander Pick, born in 1859 in Vienna, perished around 1943 in the Theresienstadt concentration camp during the Holocaust)

Pick's theorem says:

A = i + 0.5b -1

where i is the number of interior dots and b is the number of dots on the string

Apply the formula.

Rather than counting all these dots, it might be just as quick to do it directly: Consider a 12 x 13 rectangle and subtract a 1 x 8 rectangle and four right triangles.

I used Pick's Theorem, Area= i + b/2 -1 where i = dots interior to the shape, b = dot on the boundary and then subtract 1. 89 + 9 -1. Pick died in a concentration camp in the second world war, I use this theorem as an investigation for my KS3 pupils.

8 + 25 + 40 + 9 + 1 + 12 +2 = 97