Area by Coordinates Calculator
Using the Shoelace Formula (Coordinate Geometry)
Shape Type
Vertex A
Vertex B
Vertex C
Shoelace Formula
A = ½|Σ(xᵢyᵢ₊₁ - xᵢ₊₁yᵢ)|
Also known as the Surveyor's Formula
Coordinate Plane
Understanding the Shoelace Formula
The Shoelace Formula (also called the Surveyor's Formula or Gauss's Area Formula) calculates the area of any simple polygon given its vertices' coordinates.
The Formula
For a polygon with n vertices (x₁,y₁), (x₂,y₂), ..., (xₙ,yₙ):
A = ½|Σᵢ₌₁ⁿ(xᵢyᵢ₊₁ - xᵢ₊₁yᵢ)|
where (xₙ₊₁,yₙ₊₁) = (x₁,y₁) to close the polygon
For a Triangle
A = ½|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
Why "Shoelace"?
The name comes from the criss-cross pattern used when computing the formula manually, resembling the pattern of lacing a shoe.
Important Notes
- Works for any simple (non-self-intersecting) polygon
- Vertices must be in order (clockwise or counterclockwise)
- The absolute value ensures area is always positive
- Self-intersecting polygons give incorrect results
Frequently Asked Questions
Does vertex order matter?
Yes! Vertices must be listed in order as you traverse the polygon's boundary. Clockwise or counterclockwise both work, but jumping around will give wrong results.
Why use absolute value?
The formula gives a positive result for counterclockwise vertices and negative for clockwise. The absolute value ensures we always get a positive area.
Can I use negative coordinates?
Absolutely! The formula works for any coordinates in the plane, positive or negative.
What about concave polygons?
The Shoelace Formula works for concave polygons too, as long as they don't self-intersect.
Is this the same as the cross product method?
Yes! For triangles, the Shoelace Formula is equivalent to using the cross product of two sides: A = ½|AB × AC|.