Question

Find the area of the polygon with vertices at (1,3) , (7,3) ,(7,7) and (4,7)?

Answer 1

Two ways to do it.
If you plot the points, it shows a trapezium with parallel sides 6 (=7-1) and 3(=7-4) units long, and the height is 4 (=7-3).
So the area is (6+3)*4/2=18.

If you would like to calculate the area of a polygon given in corner coordinates, you can list the points in sequence like

a,b
c,d
e,f
g,h
a,b

and repeat a,b at the bottom.
Calculate the crossed sum for each successive pair of coordinates, as in:
ad-bc + cf-de + eh-fg + gb-ha
and divide the sum by 2.

The sum may be negative depending the direction of the points. So just take the absolute value.

For the given points, list:

1,3
7,3
7,7
4,7
1,3 (repeat of first point)

Products:
1*3-3*7 + 7*7-3*7 + 7*7-7*4 + 4*3-7*1
=-18 + 28 + 21 + 5
= 36
Divide by 2 to get 18.

It may look complicated, but when the figure does not happen to be a known shape, the second method is more systematic, and easier.

Answer 2

18units