Enter vertex coordinates to compute the area of any simple polygon using the Shoelace formula
Vertex 1 X
Vertex 1 Y
Vertex 2 X
Vertex 2 Y
Vertex 3 X
Vertex 3 Y
Vertex 4 X
Vertex 4 Y
Vertex 5 X
Vertex 5 Y
Result
Polygon Type
-
Area
-
Detailed Derivation
Shoelace Formula
A = (1/2) x |sum(x_i x y_{i+1} - x_{i+1} x y_i)|
Sum over i = 1 to n (vertices in order)
Vertex n+1 = Vertex 1 (closed polygon)
The Shoelace formula is a powerful method for computing the area of any simple polygon using only vertex coordinates. It works for both convex and concave polygons, making it ideal for irregular shape area calculation.
⚠Enter vertices in order (clockwise or counterclockwise). Do not repeat the first vertex at the end. Uses 5 vertices - for fewer, enter 0,0 for unused vertices.
What Is the Shoelace Formula?
The Shoelace formula is named for the diagonal pattern formed when writing the coordinates. It is also known as the Surveyors formula or Gauss area formula. It calculates polygon area as half the sum of cross products of consecutive vertex pairs.
Cross Product
Each term x_i x y_{i+1} is a directed area. Summing them gives twice the polygon area.
Concave Handling
In concave polygons, some cross products are negative (the indented section). The sum automatically adjusts.
Absolute Value
The sum may be positive (counterclockwise) or negative (clockwise). Take the absolute value and divide by 2.
Verification
Check by comparing with known shapes. A rectangle (0,0),(4,0),(4,3),(0,3) should give area = 12.
Teaching Example: Concave pentagon (0,0),(4,0),(5,3),(2,4),(0,2). Compute: (0x0)+(4x3)+(5x4)+(2x2)+(0x0)=0+12+20+4+0=36. Next sum: (0x4)+(0x5)+(3x2)+(4x0)+(2x0)=0+0+6+0+0=6. Area = (36-6)/2 = 15.
Applications
Land SurveyingGIS MappingComputer GraphicsArchitectureCAD DesignRobotics
FAQs about Polygon Area
Does the Shoelace formula work for concave polygons?▼
Yes, it works for any simple polygon (concave or convex). The cross-products automatically handle indentations.
What is vertex order?▼
List vertices in the order they appear around the boundary, either clockwise or counterclockwise. Do not cross the order.
What if I enter vertices in wrong order?▼
A crossing order gives an incorrect area. Keep vertices in sequence around the polygon boundary. Use absolute value to handle direction.
Why is it called the Shoelace formula?▼
The multiplication pattern crosses back and forth like lacing a shoe. Write coordinates in two columns and draw the diagonal lines.
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