Pick's Theorem gives a surprisingly simple way to find the area of a polygon drawn on a square lattice. Instead of splitting the shape into triangles or using coordinates in the shoelace formula, you count points on the grid. The theorem matters because it connects geometry, number patterns, and area in one clear rule.
It works for any simple lattice polygon whose vertices are all on grid points.
The formula is A = I + B/2 - 1, where I is the number of lattice points strictly inside the polygon and B is the number of lattice points on its boundary. Boundary points include every grid point along the sides, not just the vertices. Interior points must be completely inside the polygon, not on an edge.
A worked diagram usually highlights interior points in one color and boundary points in another so the counting step is visible and checkable.
Understanding Geometry: Pick's Theorem
The hardest part is usually counting the boundary correctly. A slanted side can pass through grid points that are easy to miss. To find how many equal grid intervals lie along one side, take the horizontal change and the vertical change, ignore their signs, then find their greatest common divisor.
For example, a side that moves six units right and four units up has a greatest common divisor of two. It is made of two straight lattice steps, so there is one grid point between its endpoints. A side that moves five units right and two units up has greatest common divisor one.
No extra lattice point lies on that side. Add the step counts for every side to get the full boundary count. This method avoids counting each corner twice.
The rule has a useful connection to cutting a shape into simpler pieces. A one by one grid square has no interior grid points and four boundary points. A right triangle made from half of that square has three boundary points and no interior points.
Both cases fit the area rule. More complicated lattice polygons can be divided into triangles without adding gaps or overlaps. When a cut is made along a lattice segment, the new points created on the shared cut affect the interior and boundary counts in a balanced way.
That balance is why the same area relationship survives as pieces are joined. This gives a reason the theorem works rather than making it seem like a counting trick with no structure.
Students often meet this idea when drawing shapes on graph paper, finding land areas on a grid map, or checking an answer from the shoelace method. It is especially efficient for irregular shapes with many grid points inside them. It can be slower when a shape is very large, because counting hundreds of interior points by eye invites mistakes.
In that situation, use organized rows. Count the points in each horizontal row, record the total, then check that no boundary point entered the interior count. A quick sketch with boundary points marked first makes the work much easier to audit.
Some limits matter. The boundary must form one noncrossing loop. A bow tie shape crosses itself, so its inside region is not handled by the usual theorem.
Shapes with holes need an adjusted version because the points around an inner boundary change the count differently. The grid must be a square lattice with evenly spaced horizontal and vertical lines. Finally, area is measured in square grid units.
If each grid square represents four square centimeters, first find the area in grid squares, then multiply by four. Careful labeling of units prevents a correct count from becoming an incomplete final answer.
Key Facts
- Pick's Theorem: A = I + B/2 - 1
- A is the area of the lattice polygon in square units.
- I is the number of lattice points strictly inside the polygon.
- B is the number of lattice points on the boundary, including vertices and lattice points along edges.
- The theorem applies only to simple polygons whose vertices lie on lattice points.
- For an edge from (x1, y1) to (x2, y2), the number of lattice steps on the edge is gcd(|x2 - x1|, |y2 - y1|).
Vocabulary
- Lattice point
- A point on a coordinate grid whose x-coordinate and y-coordinate are both integers.
- Lattice polygon
- A polygon whose vertices are all lattice points.
- Interior lattice point
- A lattice point that lies completely inside the polygon and not on any side.
- Boundary lattice point
- A lattice point that lies on an edge or vertex of the polygon.
- Simple polygon
- A polygon whose sides do not cross each other and that encloses one connected region.
Common Mistakes to Avoid
- Counting only the vertices as boundary points is wrong because lattice points that lie along the edges also count in B.
- Including points on an edge as interior points is wrong because interior points must be strictly inside the polygon.
- Using Pick's Theorem on a polygon with non-lattice vertices is wrong because the theorem requires every vertex to have integer coordinates.
- Forgetting the minus 1 in A = I + B/2 - 1 gives an area that is exactly 1 square unit too large.
Practice Questions
- 1 A lattice polygon has I = 7 interior lattice points and B = 10 boundary lattice points. Use Pick's Theorem to find its area.
- 2 A lattice triangle has vertices (0, 0), (6, 0), and (0, 4). Its boundary has B = 12 lattice points. Use Pick's Theorem to find the number of interior lattice points.
- 3 Explain why a square with vertices (0, 0), (2, 0), (2, 2), and (0, 2) has more boundary lattice points than just its four corners, and describe how that affects Pick's Theorem.