Pick Theorem Area Calculator
PicklatticeareaI+B/2−1, actual independently verified simple hole free integer lattice polygon counts, integer area or half integer; counts not proof valid polygon.
Description
Pick's theorem relates area to interior and boundary lattice-point counts for a simple hole-free lattice polygon.
Pick's theorem relates area to interior and boundary lattice-point counts for a simple hole-free lattice polygon.1
This tool applies those independently established counts without verifying the polygon geometry itself.
When to use Pick Theorem Area Calculator
This tool applies those independently established counts without verifying the polygon geometry itself.
How to use Pick Theorem Area Calculator and how it works
- Prepare interior lattice point count, boundary lattice point count, using the input definitions and matching units below.
- Enter the required fields and run Pick Theorem Area Calculator.
- Read result.value using the units, signs and ordering explained below.
- Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.
| Input field | Meaning | Value type | Requirement |
|---|---|---|---|
| actualInteriorLatticePointCount | Actual Interior Lattice Point Count. Use the units and matching context stated in the calculation below. | number | Required; integer 0–1000000; finite; other constraints below |
| actualBoundaryLatticePointCount | Actual Boundary Lattice Point Count. Use the units and matching context stated in the calculation below. | number | Required; integer 0–1000000; finite; other constraints below |
Require interior count I integer 0–1000000 and boundary count B integer 3–1000000. Calculate I+B/2-1 and require a positive result. Return lattice area in squared lattice-coordinate units. These checks do not prove that supplied counts come from a simple hole-free integer-lattice polygon.
latticeArea=I+B/2-1 Apply the input ordering, units and branch conditions stated above.
| Output field | Meaning | Value type |
|---|---|---|
| result | Object containing the calculated or selected fields below. | object |
| result.value | Geometric area in squared input length or lattice units under the stated convention. | number |
{
"input": {
"actualInteriorLatticePointCount": 1,
"actualBoundaryLatticePointCount": 4
},
"output": {
"result": {
"value": 2
}
}
}Limitations and assumptions
The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.
All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.
Alternative or Complementary methods
Use Torus Surface Area Calculator for a complementary operation: A ring torus has a major radius larger than its tube radius, giving a non-self-intersecting surface.
Use Star Polygon Area Calculator for a complementary operation: A regular star polygon's algebraic area counts winding multiplicity and can differ from the area of its filled union.
References
-
Pick Theorem Area Calculator — tool.io
Similar or alternative tools
- Torus Surface Area Calculator
A ring torus has a major radius larger than its tube radius, giving a non-self-intersecting surface.
- Star Polygon Area Calculator
A regular star polygon's algebraic area counts winding multiplicity and can differ from the area of its filled union.