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SimplePolygonQ[poly]

gives True if the polygon poly is simple and False otherwise.

Details
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Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Possible Issues  
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SimplePolygonQ

SimplePolygonQ[poly]

gives True if the polygon poly is simple and False otherwise.

Details

  • A polygon is simple if it has no holes and non-intersecting boundary line segments.
  • A simple polygon is topologically equivalent to a disk.

Examples

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Basic Examples  (2)

Test whether a polygon is simple:

Wolfram Language code: 𝒫 = Polygon[{{0, 0}, {1, 0}, {0.5, 0.8}}]
Wolfram Language code: SimplePolygonQ[𝒫]

SimplePolygonQ gives False for non-simple polygons:

Wolfram Language code: 𝒫 = Polygon[{{0, 0}, {1, 0}, {0.2, 1}, {0.9, 1}}]
Wolfram Language code: SimplePolygonQ[𝒫]

Scope  (5)

SimplePolygonQ works on polygons:

Wolfram Language code: 𝒫 = Polygon[{{0, 0}, {1, 0}, {0.5, 0.8}}]
Wolfram Language code: SimplePolygonQ[𝒫]

Triangle:

Wolfram Language code: SimplePolygonQ[Triangle[]]

Rectangle:

Wolfram Language code: SimplePolygonQ[Rectangle[]]

Polygon with holes:

Wolfram Language code: 𝒫 = Polygon[{{0, 0}, {3, 0}, {3, 3}, {0, 3}, {1, 1}, {1, 2}, {2, 2}, {2, 1}}, {1, 2, 3, 4} -> {{5, 6, 7, 8}}]
Wolfram Language code: SimplePolygonQ[%]

Self-intersecting polygons:

Wolfram Language code: 𝒫 = Polygon[{{0.35, 0.2}, {0.9, 0.75}, {0.1, 0.55}, {0.9, 0.35}, {0.42, 0.9}}]
Wolfram Language code: SimplePolygonQ[%]

Polygons with disconnected components:

Wolfram Language code: 𝒫 = Polygon[{{0, 0}, {1, 0}, {0, 1}, {1, 1}, {2, 1}, {1, 2}}, {{1, 2, 3}, {4, 5, 6}}]
Wolfram Language code: SimplePolygonQ[%]

Polygons in :

Wolfram Language code: Polygon[{{2, 3, 1, 0}, {6, 9, 1, 0}, {5, 4, 1, 0}, {8, 2, 1, 0}}];
Wolfram Language code: SimplePolygonQ[%]

Applications  (2)

Generate random polygons for testing algorithms and verification of time complexity:

Wolfram Language code: n = Table[Floor[10 ^ i], {i, 1, 4, 1 / 40}];
Wolfram Language code: polygons = Table[RandomPolygon[{"Simple", i}], {i, n}];
Wolfram Language code: AllTrue[polygons, SimplePolygonQ]

Time complexity for algorithms for simple polygons:

Wolfram Language code: t = First[AbsoluteTiming[SimplePolygonQ[#]]]& /@ polygons;
Wolfram Language code: Show[{ListPlot[Transpose[{n, t}], PlotRange -> All, Joined -> True, AxesLabel -> {"n", "Time"}], Plot[Style[Evaluate[Fit[Transpose[{n, t}], {1, x}, x]], Red, Dashed], {x, 0, 10 ^ 5}]}]

Polygon classification using machine learning. Train a classifier function on polygon examples:

Wolfram Language code: trainingset = Table[(RandomPolygon[{#, RandomInteger[{3, 100}]}] -> #)&@RandomChoice[{"Simple", "Convex", "StarShaped"}], {20}];
Wolfram Language code: cfunc = Classify[Image[Graphics[#1]] -> #2&@@@trainingset]

Use the classifier function to classify new polygons:

Wolfram Language code: class = cfunc[Image[Graphics[#]]]&;
Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: class[𝒫]

A simple polygon:

Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: class[𝒫]

A star‐shaped polygon:

Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: class[𝒫]

Properties & Relations  (5)

A convex polygon is simple:

Wolfram Language code: 𝒫 = Polygon[{{0, 0}, {1, 0}, {0.5, 0.8}}]
Wolfram Language code: ConvexPolygonQ[𝒫]
Wolfram Language code: SimplePolygonQ[𝒫]

The OuterPolygon of a simple polygon is simple:

Wolfram Language code: 𝒫 = Polygon[{{0, 0}, {1, 0}, {0.5, 0.8}}]
Wolfram Language code: SimplePolygonQ[𝒫]
Wolfram Language code: SimplePolygonQ[OuterPolygon[𝒫]]

Simple polygons do not have holes:

Wolfram Language code: InnerPolygon[𝒫]

Use PolygonDecomposition to decompose a polygon into simple polygons:

Wolfram Language code: 𝒫 = Polygon[{{0, 0}, {3, 0}, {3, 3}, {0, 3}, {1, 1}, {1, 2}, {2, 2}, {2, 1}}, {1, 2, 3, 4} -> {{5, 6, 7, 8}}];
Wolfram Language code: PolygonDecomposition[𝒫, "Simple"]
Wolfram Language code: Graphics[{EdgeForm[Gray], %}]

Use RandomPolygon to generate a simple polygon:

Wolfram Language code: RandomPolygon["Simple"]
Wolfram Language code: SimplePolygonQ[%]

The number of edges of a simple polygon always equals the number of vertices:

Wolfram Language code: 𝒫 = DynamicModule[«3»];
Wolfram Language code: MeshCellCount[𝒫, 0] == MeshCellCount[𝒫, 1]

Possible Issues  (1)

For nonconstant polygons, SimplePolygonQ returns False:

Wolfram Language code: 𝒫 = Polygon[{{a, b}, {c, d}, {e, f}, {g, h}}];
Wolfram Language code: ConstantRegionQ[𝒫]
Wolfram Language code: SimplePolygonQ[𝒫]
Wolfram Research (2019), SimplePolygonQ, Wolfram Language function, https://reference.wolfram.com/language/ref/SimplePolygonQ.html.

Text

Wolfram Research (2019), SimplePolygonQ, Wolfram Language function, https://reference.wolfram.com/language/ref/SimplePolygonQ.html.

CMS

Wolfram Language. 2019. "SimplePolygonQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SimplePolygonQ.html.

APA

Wolfram Language. (2019). SimplePolygonQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SimplePolygonQ.html

BibTeX

@misc{reference.wolfram_2026_simplepolygonq, author="Wolfram Research", title="{SimplePolygonQ}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/SimplePolygonQ.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_simplepolygonq, organization={Wolfram Research}, title={SimplePolygonQ}, year={2019}, url={https://reference.wolfram.com/language/ref/SimplePolygonQ.html}, note=[Accessed: 13-June-2026]}