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```Date: 04/29/2002 at 18:12:27
From: Aaron DeNOsaquo

Given the (x,y) coordinates of four points, is there a simple formula
to compute the area of the quadrilateral with corners on these 4
points?
```

```
Date: 04/29/2002 at 18:29:39
From: Doctor Tom

Hi Aaron,

Yes.

If the points are (x1, y1), (x2, y2), ..., (x4, y4), then here's the
formula:

2A = (x1y2 - x2y1) + (x2y3 - x3y2) + (x3y4 - x4y3) + (x4y1 - x1y4)

(Notice the formula is for 2 times the area - to get the area,
calculate the number on the right and divide by 2.)

Also, the number you get will depend on whether you go clockwise or
counter-clockwise around the quadrilateral. One direction will give
the negative of the other, so to get the actual area, find the area as
above and then take the absolute value.

If the four points connecting 1 to 2 to 3 to 4 to 1 have lines that
cross each other, the answer is meaningless - they have to form a
simple region with an inside and an outside and no crossing lines.
The formula does work for non-convex quadrilaterals, however.
"Non-convex" means that there can be "indentations."

Also, I hope that the pattern of the terms above is obvious. The same
formula (but with more or fewer terms) will work for any simple
polygon, from triangle on up. (A "simple" polygon is one whose edges
do not cross each other.)

- Doctor Tom, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 04/29/2002 at 18:32:52
From: Aaron DeNOsaquo

I just wanted to send thanks for the quick response. I am a grad
student and am working on a finite element program. This will really
help. Thanks again.
```
Associated Topics:
High School Coordinate Plane Geometry
High School Triangles and Other Polygons

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