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Pick's Theorem

Tool to apply and calculate a surface using the Pick's Theorem that allows the calculation of the area of a polygon positioned on a lattice (normalized orthogonal grid) and whose vertices are points of the grid.

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Pick's Theorem -

Tag(s) : Geometry

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# Pick's Theorem

## Pick Polygon Area Calculator

### What is the Pick Theorem? (Definition)

Pick's Theorem is a mathematical formula that allows you to calculate the area of a polygon whose vertices have integer coordinates in a Cartesian plane (a 2D grid/grid).

For a polygon with $b$ vertices constructed on a grid (the vertices are grid points) having $i$ points inside it, Pick's formula indicates that the area $A$ of the polygon is $$A = i + \frac{b}{2} - 1$$

All points present on the contour are considered vertices (vertex angles are flat in this case).

### How to calculate an area with the Pick Theorem?

Pick's formula requires two parameters: the number $i$ of interior points of the polygon and the number $b$ of points on the contour/perimeter of the polygon. The area $A$ of the polygon is $A = i + \frac{b}{2} - 1$

Example: The polygon drawn below has 15 points inside the polygon (light gray), and 10 vertices (dark gray). Its area is therefore $A = 15 + 10/2 - 1 = 19$.

### Does Pick's Theorem work for concave polygons?

Yes, Pick's Theorem can be applied to concave polygons as long as the vertices have integer coordinates.

### How to use Pick's Theorem for polygons with holes inside?

Pick's Theorem only applies to simple polygons with no holes inside. However, if the hole(s) form a polygon eligible for Pick's Theorem, then it is possible to calculate the area of the hole polygon (ignoring the holes) and subtract the areas of the holes to obtain the area of the hole. polygon with holes.

### Who created Pick's Theorem?

The formula owes its name to Georg Alexander Pick who described it in 1899.

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Pick's Theorem on dCode.fr [online website], retrieved on 2024-06-15, https://www.dcode.fr/pick-theorem

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