Star Polygon Area Calculator
Regulars tar{n/k}ALGEBRAIC signed shoelace area with winding multiplicity nR²sin(2πk/n)/2, not union filled area; primitive gcd(n, k)=1and1≤k<n/2, n≥5.
Description
A regular star polygon's algebraic area counts winding multiplicity and can differ from the area of its filled union.
A regular star polygon's algebraic area counts winding multiplicity and can differ from the area of its filled union.1
This tool calculates that algebraic area for a primitive coprime vertex-step convention.
When to use Star Polygon Area Calculator
This tool calculates that algebraic area for a primitive coprime vertex-step convention.
How to use Star Polygon Area Calculator and how it works
- Prepare vertex count, circumradius, step, using the input definitions and matching units below.
- Enter the required fields and run Star Polygon 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 |
|---|---|---|---|
| actualVertexCount | Integer number of vertices n in the primitive polygon convention. | number | Required; integer 1–1000000; finite; other constraints below |
| actualCircumradius | Actual Circumradius. Use the units and matching context stated in the calculation below. | number | Required; strictly positive; finite; other constraints below |
| actualStep | Integer traversal step k; coprime to n and smaller than n/2. | number | Required; integer 1–1000000; finite; other constraints below |
Require vertex count n integer at least 5 and at most 1000000, positive circumradius R, and integer step k with 1<=k<n/2. Compute gcd(n,k) and reject nonprimitive cases where it is not one. Return n*R^2*sin(2*pi*k/n)/2 in squared length units as algebraic shoelace area with winding multiplicity, not union-filled area. The branch accepts k=1, yielding a regular polygon.
algebraicArea=n*R^2*sin(2*pi*k/n)/2 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": {
"actualVertexCount": 5,
"actualCircumradius": 1,
"actualStep": 2
},
"output": {
"result": {
"value": 1.4694631307311832
}
}
}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 Regular N-Gon Perimeter Calculator for a complementary operation: A regular polygon perimeter is its number of equal sides times their common length.
References
-
Star Polygon 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.
- Regular N-Gon Perimeter Calculator
A regular polygon perimeter is its number of equal sides times their common length.