Lattice points are points on a coordinate grid whose coordinates are both integers. Pick's Theorem gives a fast way to find the area of certain polygons drawn on a lattice. This cheat sheet helps students count boundary and interior lattice points accurately, then connect those counts to area.
It is useful for geometry, coordinate geometry, contest math, and visual problem solving.
The key formula is Pick's Theorem, , where is area, is the number of interior lattice points, and is the number of boundary lattice points. For a line segment between lattice points, the number of lattice points on the segment is related to . Students should understand when the theorem applies, how to count points without double-counting vertices, and how to check answers using area formulas such as the shoelace formula.
Key Facts
- A lattice point is a point where both and are integers.
- Pick's Theorem states that the area of a simple lattice polygon is .
- In Pick's Theorem, is the number of lattice points strictly inside the polygon and is the number of lattice points on the boundary.
- For a segment from to , the number of boundary lattice points on the segment including both endpoints is .
- When adding boundary counts around a polygon, use so vertices are not double-counted.
- Pick's Theorem can be rearranged to find interior points: .
- Pick's Theorem can be rearranged to find boundary points: .
- Pick's Theorem applies only to simple polygons whose vertices are lattice points, not to curves, self-intersecting polygons, or polygons with non-lattice vertices.
Vocabulary
- Lattice point
- A point on the coordinate plane with integer coordinates, such as .
- Lattice polygon
- A polygon whose vertices are all lattice points.
- Boundary point
- A lattice point that lies on an edge or vertex of the polygon.
- Interior point
- A lattice point that lies strictly inside the polygon and not on any edge.
- Pick's Theorem
- A formula for the area of a simple lattice polygon, given by .
- Greatest common divisor
- The greatest common divisor, written , is the largest positive integer that divides both and .
Common Mistakes to Avoid
- Counting vertices twice is wrong because each vertex belongs to two edges. Use when counting boundary points around a polygon.
- Including boundary points as interior points is wrong because Pick's Theorem separates them into and . Points on an edge or vertex must be counted in , not .
- Using Pick's Theorem on a polygon with non-integer vertices is wrong because the theorem requires every vertex to be a lattice point. Check that each vertex has coordinates with integer and integer .
- Forgetting the in is wrong because it changes the area by one square unit. Always write the full formula before substituting values.
- Using to count points on a slanted segment is wrong because diagonal lattice spacing depends on the greatest common divisor. Use for one segment including endpoints.
Practice Questions
- 1 A lattice polygon has interior lattice points and boundary lattice points. Find its area using .
- 2 Find the number of lattice points on the segment from to , including both endpoints.
- 3 A simple lattice polygon has area and boundary lattice points. Find the number of interior lattice points.
- 4 Explain why Pick's Theorem cannot be used directly for a triangle with vertices , , and .
Understanding Lattice Points & Pick's Theorem
The boundary count often causes the most mistakes. A slanted side can pass through extra grid intersections even when it does not look horizontal or vertical. The pattern comes from the horizontal and vertical changes along that side.
If a segment moves six units right and four units up, the greatest common divisor of six and four is two. This means the segment is split into two equal steps between lattice points, so there is one lattice point between its endpoints.
A move of six right and four up has the same lattice pattern as a move of three right and two up, repeated twice. The greatest common divisor detects that repeated pattern.
When counting a whole polygon, treat each side as a path that starts at one vertex and stops just before counting that vertex again on the next side. This is why adding the greatest common divisor for every side works so neatly. A rectangle that is four units wide and three units high has boundary count fourteen, not eighteen.
Its two horizontal sides contribute four each, and its two vertical sides contribute three each. The corner points belong to two sides in a sketch, but the side-by-side counting method gives each corner only one place in the final total. Drawing a small dot on every boundary point is a good check for unfamiliar shapes.
Pick's Theorem is especially useful because it turns a shape problem into a counting problem. It works for concave polygons as well as convex ones. An inward dent does not break the rule, provided the edges do not cross.
This matters in coordinate geometry because irregular shapes can be awkward to split into rectangles and triangles. The theorem can reveal missing information too. If the area and boundary count are known, the number of points inside is forced.
Since half of the boundary count appears in the calculation, an odd boundary count leads to an area ending in one half. That gives a quick reasonableness check. A result with an impossible sign or an unexpected fractional part usually signals a counting error.
Students meet these ideas in grid maps, pixel art, computer graphics, land plots, and math contest diagrams. The grid gives a discrete model, meaning only certain locations count as points. Area is still continuous, so a polygon may have a whole-number area or a half-unit area.
Keep the two ideas separate. Count dots for boundary and interior information, then use square units for area. For practice, begin with rectangles and right triangles, where area can be checked directly by base times height or one half times base times height.
Then try tilted shapes and concave shapes. Always check that every corner has integer coordinates, trace the boundary once without crossing it, and verify the final area by breaking the figure into simpler pieces when possible.