# How to calculate the area of a polygon? [duplicate]

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How quickly we forget - basic trig. Calculate the area of a polygon

For a program I want to calculate the area of a given polygon. It may have every form, the sides even may cross. Do you can give me pointers to a way to calculate areas for such polygons?

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## marked as duplicate by joriki, J. M., Asaf Karagila, t.b., Zev ChonolesDec 24 '11 at 18:59

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How is the polygon given? If its boundary is not a simply closed curve you will have to specify what is meant by "area". – Christian Blatter Sep 22 '11 at 13:42
It is given as a closed linestring, but the lines can cross. – Mnementh Sep 22 '11 at 13:58
it would be worth looking at this question: math.stackexchange.com/questions/15739/… – yasmar Sep 22 '11 at 18:56
See the Wikipedia entry on this topic: Polygon area – Joseph O'Rourke Sep 22 '11 at 22:58

## 1 Answer

Let us assume that polygon P has vertices (x1,y1), ..., (xn,yn). Then we can represent the area of the polygon as a sum of trapezoids T(1,2)=(x1,y1)-(x1,0)-(x2,0)-(x2,y2), ..., T(n,1), where area can be counted with opposite sign if x2 < x1:

area(P) = area(T(1,2)) + area(T(2,3)) + ... + area(T(n,1))

and obviously

area(T(i,j)) = (xj-xi)*(yj+yi)/2

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Does that really work if the sides are crossing? Like a polygon representing an eight? – Mnementh Sep 22 '11 at 12:50