Pick's Theorem:

Pick's Theorem:

Pick's Theorem provides a simple way to calculate the area of a polygon whose vertices are lattice points (points with integer coordinates) on a grid. 

The formula is: Area = I + B/2 - 1, 

where 'I' is the number of lattice points in the interior of the polygon, and 'B' is the number of lattice points on the boundary of the polygon. 

Explanation:

Lattice Points: These are points on a grid with integer coordinates (e.g., (1, 2), (3, -1), (0, 0)). 

Interior Points (I): These are the points that lie strictly inside the polygon. 

Boundary Points (B): These are the points that lie on the edges of the polygon. 

Formula: The theorem states that the area of the polygon is equal to the number of interior points plus half the number of boundary points, minus 1. 

Example:

Let's say a polygon has 7 interior points (I = 7) and 10 boundary points (B = 10). 

Using Pick's Theorem:

Area = 7 + (10/2) – 1 

          = 7 + 5 – 1 

           = 11

The area of the polygon is 11 square units. 

Key Points:

Pick's Theorem only applies to simple polygons (no self-intersections) with vertices on a lattice. 

It provides a surprisingly simple way to calculate area without relying on more complex geometric formulas. 

The theorem was published by Georg Alexander Pick in 1899.