Imagine a sheet of graph paper, marked with dots wherever two lines cross. Draw any polygon you like on it, taking care that every corner of the polygon lands exactly on one of the dots. Now, instead of using the geometry-class formulas for area — base times height, the shoelace rule, integration — try something simpler. Count the dots strictly inside the polygon, count the dots on its edges, and combine the two numbers with one short rule. The result is the polygon’s exact area, every time, no matter how oddly shaped.
This is Pick’s theorem, proved by Georg Pick in 1899. It is one of those mathematical statements that feels almost like a magic trick the first time you see it work, and the trick has a clean and beautiful explanation. This article is about both — the theorem and the picture behind it.
The statement
Let be a simple polygon — one whose edges do not cross — and suppose all its vertices sit on lattice points, that is, points with integer coordinates. Write for the number of lattice points lying strictly inside , and for the number of lattice points lying on the boundary of (corners and edge points alike). Then the area of is
Three small ingredients — two whole-number counts and a single constant — and they give an exact answer for an area that could be a half, a quarter, a perfectly horrid fraction, anything at all. The recipe never fails for a simple lattice polygon.
A worked example
The polygon below has vertices at — an irregular pentagon. The lattice points on its boundary are coloured red and the ones strictly inside are coloured green; the surrounding lattice dots are shown faintly for reference.
Count carefully: there are green dots inside and red dots on the boundary. Pick’s formula gives
A direct calculation using the shoelace formula confirms it: the area really is exactly . No measurement, no calculus — just two counts and an arithmetic.
Why it works
The honest proof has two moves, and the second is delicate, but the spirit is captured in a few sentences.
First, the formula is additive. Suppose you glue two lattice polygons together along a shared edge. The interior points of the joined shape are: all the original interior points of both pieces, plus the points that used to be on the shared edge (those are now in the interior). When you add up the right-hand side of Pick’s formula for the two pieces, the boundary points on the shared edge get counted twice — once at each, summing to one — but those points should each contribute a full as interior points of the joined shape. The bookkeeping works out exactly: the formula behaves correctly under gluing, provided you also handle the constant carefully, which subtracts one too many for the combined shape and gets corrected by the two boundary endpoints of the shared edge becoming interior.
Second, the formula is true for triangles whose only lattice points are their three corners. Such “empty” triangles — called primitive triangles — turn out to have area exactly . For one of them, , , and Pick’s formula gives , the correct answer. This is a nontrivial fact in its own right, equivalent to a basic result in the geometry of numbers, but once accepted it does the heavy lifting.
These two pieces fit together. Any lattice polygon can be carved up into primitive triangles, like cutting a pizza into the smallest possible slices. The formula is correct on each slice, and additive when the slices are glued back together, so it is correct on the whole. That is the modern outline of the proof Pick gave in 1899.
A second way of thinking about it makes the less mysterious. Each lattice point can be assigned a “share” of area: a point well inside the polygon owns a full unit cell of area , a point on a straight edge owns half a unit cell since the cell is split between inside and outside, and a point at a corner owns a fraction depending on the angle. Adding up all the shares gives the area. For a simple polygon the corner angles sum to , and the total deficit from the boundary points — half-units plus angle corrections — works out to . The is the total turning, the same constant that appears in the Euler characteristic of a closed disk.
A small theorem with a long reach
Pick’s theorem is a stunning example of how counting and geometry intersect. It belongs to a wider tradition in mathematics — counting lattice points in shapes — that runs from Gauss’s circle problem (how many lattice points sit inside a circle of given radius) through the modern theory of Ehrhart polynomials, which count the lattice points inside the dilated copies of a fixed polygon and produce polynomial formulas in that encode geometric data far beyond area.
The practical applications are more down-to-earth. In digital image processing and computer graphics, areas of regions defined by pixels are essentially Pick’s counts. Geographic software that lets a user trace a region on a map with a snap-to-grid tool uses essentially the same idea. And as a piece of pure mathematical pleasure, the theorem remains hard to beat: every time you draw a new lattice polygon, the formula reaches out and tells you its area before your pencil has stopped moving. A small theorem, but one with the unmistakable feel of a permanent truth.
Frequently asked
Who was Pick and when did he prove this?
Georg Alexander Pick was an Austrian mathematician who published this result in 1899. He worked in Prague and is known mainly for this theorem, though it sat in relative obscurity for decades before being rediscovered and popularised by Hugo Steinhaus in the 1950s. Pick himself died in the Theresienstadt concentration camp in 1942.
Does Pick's theorem work for any polygon?
It works for any simple polygon — meaning a polygon whose edges do not cross themselves — whose vertices all sit on lattice points (points with integer coordinates). The polygon may be wildly irregular and have any number of sides. It does not work for polygons with curved sides, with vertices off the lattice, or with holes inside (though there is a generalisation with a small correction term for each hole).
Why does the formula have a −1?
It comes from the topology of a closed polygon. The intuitive picture is that each interior lattice point contributes a full unit of area, each boundary lattice point contributes half a unit (since it sits on an edge shared between inside and outside), and the polygon as a whole, taken once around, contributes the missing turning angle that adds up to one full revolution — which subtracts a single unit. The constant is exactly the Euler characteristic of a disk in disguise.
What is Pick's theorem useful for?
Beyond its charm as a counting trick, it appears in digital geometry — computing pixel areas in images — and in software that estimates the area of a shape drawn on a grid, such as map tools and CAD systems with snap-to-grid. It is also the seed of a large field called Ehrhart theory, which counts lattice points inside scaled-up polygons and underlies surprising connections between combinatorics, algebraic geometry, and number theory.