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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?

Pick's theorem (or rule) easily calculates the area (surface) of a polygon with $b$ vertices built on a lattice, a 2D grid of points with integer coordinates (points with equal distances). If all $b$ vertices of the polygon (vertices can be flat) are grid points and the polygon has $i$ points inside itself then Pick's formula indicates that the polygon area $A$ is equal to $$A = i + \frac{b}{2} - 1$$

All the points present on the contour are considered as vertices.

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

The Pick formula requires only two parameters: the number $i$ of interior points of the polygon and the number $b$ of vertices of the polygon (which is in the number of grid points on the perimeter of the polygon). Thearea $A$ 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 $A = 15 + 10/2 - 1 = 19$.

### Who created Pick's Theorem?

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

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