Geometric figures in the square grid
Maths circle meeting on 14 June 2025
Room and time: virtual/online with the video conferencing software Zoom (Paderborn University campus licence) from 10:00 to 13:00
Leader of the workshop: Dr Kerstin Hesse
Description: We consider a rectangular square grid in which all points of the grid have a distance of 1 to the direct link to the neighbouring points. The grid points are therefore (m,n), where m, n are integers. How do you calculate the area A of a polygon whose vertices are grid points? Pick's theorem states that this can be done using the formula A = i + (1/2)*r - 1, where i is the number of grid points in the centre of the polygon and r is the number of grid points on the edge of the polygon! So you only need to be able to count grid points to calculate the area. - Why does this surprising theorem apply and how do you prove it? We will approach the proof step by step by first proving Pick's theorem for rectangles and right-angled axis-parallel triangles, each with vertices in grid points. An extension to the case of arbitrary triangles with vertices in grid points and the additivity/subtractivity of the area calculation provides the proof for arbitrary polygons with vertices in grid points.
