BTT Mathematics / Primary Mathematics Learning Hub / Lattice Points and Coordinate Area
A lattice point is a point with whole-number coordinates. When a simple polygon has all its vertices on lattice points, its area can be connected to the number of lattice points strictly inside the polygon and the number lying on its boundary.
Pick’s theorem states that for a simple lattice polygon—a non-self-intersecting polygon whose vertices lie on lattice points—
Area = 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, including vertices.
For a 4-by-3 lattice rectangle, area is 12. Its boundary contains 14 lattice points and its interior contains 6. Check: 6+14/2−1=6+7−1=12.
This guide is Primary Mathematics enrichment. It connects coordinates, area, counting and proof. It extends Geometry, Angles, Symmetry and Coordinates, Triangle Area and Composite Figures and Counting Rectangles, Squares and Triangles.
Pick’s theorem is optional enrichment, not a claim about universal Primary syllabus requirements. Use the MOE Primary curriculum page and the learner’s school programme for required scope.
Lattice points · Count boundary points · Use Pick’s theorem · Coordinate area · When the theorem does not apply · 24 questions · Worked answers · Teaching and transfer
1. Start by separating interior, boundary and exterior points
On a square grid, lattice points are grid intersections with integer coordinates such as (0,0), (3,−2) and (7,5). A point may be a polygon vertex, lie on an edge between vertices, lie strictly inside the polygon or lie outside it.
Worked example A: A 3-by-2 rectangle
Take vertices (0,0),(3,0),(3,2),(0,2). Area=6. Boundary lattice points are the points around the outside: B=10. The interior points are (1,1) and (2,1), so I=2. Pick’s theorem gives 2+10/2−1=6.
Worked example B: A 6-by-6 square
Area=36. Boundary B=4×6=24 because each side contributes six lattice intervals and the four vertices are handled once by the full boundary count. Interior lattice points form a 5-by-5 array, so I=25. Check: 25+12−1=36.
Worked example C: A right triangle
Triangle vertices (0,0),(4,0),(0,3) have area 1/2×4×3=6. The boundary has 8 lattice points. Pick gives I=6−8/2+1=3.
Interior means strictly inside
A lattice point on an edge is counted in B, not I. A vertex is also part of B. Double-counting a boundary point as interior breaks the formula.
2. Boundary counting follows lattice steps along each edge
For a horizontal lattice edge of length 7 units, there are eight lattice points if both endpoints are counted. But when counting the boundary of a closed polygon edge by edge, a convenient rule is to add the number of lattice steps on each edge so shared vertices are not double-counted.
For an edge whose coordinate change is (Δx,Δy), the number of lattice steps along that edge is gcd(|Δx|,|Δy|). Summing this value over all polygon edges gives B.
Worked example D: Sloping segment
From (0,0) to (6,4), gcd(6,4)=2. The segment has two equal lattice steps and therefore three lattice points including endpoints: (0,0),(3,2),(6,4). It has one interior lattice point on the segment.
Worked example E: Primitive segment
From (0,0) to (5,2), gcd(5,2)=1. There is no lattice point strictly between the endpoints. Such a segment is sometimes called primitive.
Worked example F: Triangle boundary
For triangle (0,0),(6,0),(0,4): edge step counts are gcd(6,0)=6, gcd(6,4)=2 and gcd(0,4)=4. Hence B=12.
Worked example G: Rectangle shortcut
An axis-aligned a-by-b lattice rectangle has boundary count B=2a+2b. A 7-by-3 rectangle therefore has B=20.
3. Pick’s theorem can find area, interior count or boundary count
The equation A=I+B/2−1 can be rearranged according to the unknown quantity.
Worked example H: Find interior points
A simple lattice polygon has area 15 and B=12. Then I=15−6+1=10.
Worked example I: Find area
If I=5 and B=10, area=5+5−1=9.
Worked example J: Half-unit area
If I=7 and B=7, area=7+3.5−1=9.5 square units. Lattice polygons can have half-integer area.
Worked example K: No interior points
If I=0 and B=4, area=0+2−1=1. A simple lattice polygon may enclose area without containing any interior lattice point.
Worked example L: A larger right triangle
Triangle (0,0),(8,0),(0,6) has area 24. Boundary B=8+gcd(8,6)+6=8+2+6=16. Thus I=24−8+1=17.
Worked example M: Another triangle
Triangle (0,0),(6,0),(2,4) has area 12. Its boundary count is gcd(6,0)+gcd(4,4)+gcd(2,4)=6+4+2=12. So I=12−6+1=7.
4. Coordinate area gives an independent check
Pick’s theorem should not become a replacement for understanding ordinary area. For rectangles and right triangles, compute the area directly from lengths and use Pick’s theorem as a second representation.
Worked example N: Rectangle from coordinates
Vertices (2,1),(7,1),(7,5),(2,5) form a 5-by-4 rectangle. Direct area=20. Boundary B=18 and interior I=(5−1)(4−1)=12. Pick gives 12+9−1=20.
Worked example O: Right triangle from coordinates
Vertices (1,2),(8,2),(1,7) have perpendicular base seven and height five. Area=17.5. Boundary B=7+gcd(7,5)+5=7+1+5=13, so I=17.5−6.5+1=12.
Worked example P: Translation preserves all counts
Moving every vertex of a lattice polygon by the same integer vector preserves edge differences, area, B and I. A translated 4-by-3 rectangle still has area 12, B=14 and I=6.
Worked example Q: Scaling changes the counts
Doubling every coordinate of a 3-by-2 rectangle produces a 6-by-4 rectangle. Area scales by four from 6 to 24. Boundary changes from 10 to 20, and interior changes from 2 to 15. Pick verifies 15+10−1=24.
5. The theorem has conditions; do not use it blindly
Pick’s theorem in this guide applies to simple polygons whose vertices are lattice points. If an edge crosses another non-adjacent edge, the polygon is self-intersecting and the ordinary simple-polygon form should not be applied directly.
Non-lattice vertices
A triangle with vertex (1/2,2) is not a lattice polygon under the present definition. Ordinary coordinate geometry may still find its area, but the simple Pick formula is not justified.
Curved boundaries
A circle drawn through lattice points is not a polygonal boundary. Pick’s theorem is not a circle-area formula.
Holes need extra care
A region with holes requires a generalized version with topology corrections. This guide restricts practice to simple polygons without holes.
Check the inventory
Before applying the formula, verify: vertices are lattice points; boundary is a simple closed polygon; B includes every boundary lattice point once; I includes only strict interior points.
6. Practice: 24 original questions
Use A=I+B/2−1 only for the simple lattice polygons described.
Questions 1–8: Direct Pick calculations
1. A=12 and B=14. Find I.
2. A=6 and B=8. Find I.
3. I=5 and B=10. Find A.
4. I=0 and B=6. Find A.
5. A 5-by-4 lattice rectangle: find A, B and I.
6. A square of side 6 on the lattice: find A, B and I.
7. Triangle (0,0),(6,0),(0,4): find B and I.
8. A 3-by-2 lattice rectangle: find B and I.
Questions 9–16: Boundary lattice points
9. Find B for a 7-by-3 lattice rectangle.
10. Find B for a square of side 5.
11. Triangle (0,0),(5,0),(0,5): find B and I.
12. Triangle (0,0),(8,0),(0,6): find B and I.
13. How many lattice points, including endpoints, lie on the segment from (0,0) to (6,4)?
14. How many lattice points, including endpoints, lie on the segment from (1,1) to (7,5)?
15. Does the segment from (0,0) to (5,2) contain a lattice point strictly between its endpoints?
16. Find B for rectangle (0,0),(6,0),(6,4),(0,4).
Questions 17–24: Coordinate area and theorem limits
17. I=8 and B=10. Find A.
18. I=12 and B=8. Find A.
19. A=15 and B=12. Find I.
20. A=9.5 and B=7. Find I.
21. Triangle (0,0),(7,0),(0,5): find A, B and I.
22. Triangle (0,0),(6,0),(2,4): find A, B and I.
23. Can the simple-polygon form of Pick’s theorem be applied directly to a self-intersecting bow-tie polygon? Explain.
24. I=0 and B=4. Find A.
7. Worked answers
Answers 1–8
1. I=6. 12−7+1.
2. I=3. 6−4+1.
3. A=9. 5+5−1.
4. A=2. 0+3−1.
5. A=20, B=18, I=12. I=20−9+1.
6. A=36, B=24, I=25.
7. B=12, I=7. Area 12; I=12−6+1.
8. B=10, I=2.
Answers 9–16
9. B=20. 2(7+3).
10. B=20. Four sides each contribute five lattice steps.
11. B=15, I=6. Area=12.5; B=5+5+5=15; I=12.5−7.5+1.
12. B=16, I=17. Area=24; B=8+2+6.
13. 3 lattice points. gcd(6,4)=2 gives two lattice steps and therefore three points including endpoints.
14. 3 lattice points. Coordinate difference is (6,4), again gcd 2.
15. No. gcd(5,2)=1, so only the endpoints are lattice points on the segment.
16. B=20. 2(6+4).
Answers 17–24
17. A=12. 8+5−1.
18. A=15. 12+4−1.
19. I=10. 15−6+1.
20. I=7. 9.5−3.5+1.
21. A=17.5, B=13, I=12. B=7+1+5.
22. A=12, B=12, I=7. B=6+4+2.
23. No. The version used here requires a simple non-self-intersecting lattice polygon.
24. A=1. 0+2−1.
8. Teaching and transfer
If a learner uses the formula mechanically, first count a rectangle directly. Area, boundary points and interior points should all be visible. Then show that the formula compresses a relationship that the grid already displays.
When boundary points are double-counted
Trace the boundary once in one direction. For sloping edges, count lattice steps using gcd rather than adding endpoint counts from every edge and accidentally repeating vertices.
When interior and boundary are mixed
Mark boundary lattice points with one symbol and strict interior points with another. The theorem keeps these populations separate.
When a half-unit area feels suspicious
Use the right triangle with legs five and five. Its area is 12.5, while every vertex remains on the lattice. Lattice geometry naturally includes half-square areas.
When the theorem is applied outside its conditions
Ask for a checklist before substitution: lattice vertices? simple polygon? no holes? correct B count? The theorem is powerful precisely because its conditions are explicit.
Connect to invariants and counting
Coordinate area, boundary-step counts and interior counts offer independent checks. This makes the guide a bridge between Geometric Dissections and Area Proofs and systematic counting.
Continue through this enrichment collection
For reflection through physical folds, use Paper Folding, Crease Patterns and Symmetry. For remainder-based gcd reduction, use Euclidean Algorithm, GCD and Remainder Reduction. For ordered fraction neighbours, use Farey Sequences, Mediants and Fraction Neighbours.
Return to the BTT Primary Mathematics Learning Hub.
Original enrichment guide with 24 original practice questions and separate worked answers. Pick’s theorem is restricted here to simple lattice polygons without holes.