Area from Coordinates (Shoelace Formula)
Area from Coordinates (Shoelace Formula)
\[A=\frac{\left|x_1y_2+x_2y_3+x_3y_4+x_4y_1-y_1x_2-y_2x_3-y_3x_4-y_4x_1\right|}{2}\]
Variables
A = polygon area
x1 = x-coordinate of first vertex
y1 = y-coordinate of first vertex
x2 = x-coordinate of second vertex
y2 = y-coordinate of second vertex
x3 = x-coordinate of third vertex
y3 = y-coordinate of third vertex
x4 = x-coordinate of fourth vertex
y4 = y-coordinate of fourth vertex
Description
What is this formula?
This formula calculates the area of a polygon from the coordinates of its vertices using the Shoelace Formula.
When to use it
Use this formula in coordinate geometry, surveying, GIS, engineering, and computer graphics when polygon vertices are known.
Example
For the vertices:
(1,1)
(5,1)
(4,4)
(1,3)
A = abs((1*1+5*4+4*3+1*1-1*5-1*4-4*1-3*1)/2)
A = 10.5
Applications
- Land surveying
- GIS and mapping
- CAD and computer graphics
- Coordinate geometry
- Engineering calculations
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