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- Pick's_theorem abstract "Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the number b of lattice points on the boundary placed on the polygon's perimeter:In the example shown, we have i = 7 interior points and b = 8 boundary points, so the area is A = 7 + 8/2 − 1 = 7 + 4 − 1 = 10 (square units)Note that the theorem as stated above is only valid for simple polygons, i.e., ones that consist of a single piece and do not contain "holes". For a polygon that has h holes, with a boundary in the form of h + 1 simple closed curves, the slightly more complicated formula i + b/2 + h − 1 gives the area.The result was first described by Georg Alexander Pick in 1899.The Reeve tetrahedron shows that there is no analogue of Pick's theorem in three dimensions that expresses the volume of a polytope by counting its interior and boundary points. However, there is a generalization in higher dimensions via Ehrhart polynomials. The formula also generalizes to surfaces of polyhedra.".
- Pick's_theorem thumbnail Pick-theorem.png?width=300.
- Pick's_theorem wikiPageExternalLink PicksTheorem.
- Pick's_theorem wikiPageExternalLink Pick.shtml.
- Pick's_theorem wikiPageExternalLink pick.pdf.
- Pick's_theorem wikiPageExternalLink Pick.html.
- Pick's_theorem wikiPageID "328252".
- Pick's_theorem wikiPageRevisionID "588037365".
- Pick's_theorem hasPhotoCollection Pick's_theorem.
- Pick's_theorem title "Pick's Theorem".
- Pick's_theorem urlname "PicksTheorem".
- Pick's_theorem subject Category:Analytic_geometry.
- Pick's_theorem subject Category:Area.
- Pick's_theorem subject Category:Articles_containing_proofs.
- Pick's_theorem subject Category:Digital_geometry.
- Pick's_theorem subject Category:Euclidean_plane_geometry.
- Pick's_theorem subject Category:Lattice_points.
- Pick's_theorem subject Category:Polygons.
- Pick's_theorem subject Category:Theorems_in_geometry.
- Pick's_theorem type Abstraction100002137.
- Pick's_theorem type Attribute100024264.
- Pick's_theorem type Communication100033020.
- Pick's_theorem type Figure113862780.
- Pick's_theorem type Message106598915.
- Pick's_theorem type PlaneFigure113863186.
- Pick's_theorem type Polygon113866144.
- Pick's_theorem type Polygons.
- Pick's_theorem type Proposition106750804.
- Pick's_theorem type Shape100027807.
- Pick's_theorem type Statement106722453.
- Pick's_theorem type Theorem106752293.
- Pick's_theorem type TheoremsInGeometry.
- Pick's_theorem comment "Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the number b of lattice points on the boundary placed on the polygon's perimeter:In the example shown, we have i = 7 interior points and b = 8 boundary points, so the area is A = 7 + 8/2 − 1 = 7 + 4 − 1 = 10 (square units)Note that the theorem as stated above is only valid for simple polygons, i.e., ones that consist of a single piece and do not contain "holes". ".
- Pick's_theorem label "Formule van Pick".
- Pick's_theorem label "Pick's theorem".
- Pick's_theorem label "Satz von Pick".
- Pick's_theorem label "Teorema de Pick".
- Pick's_theorem label "Teorema de Pick".
- Pick's_theorem label "Teorema di Pick".
- Pick's_theorem label "Théorème de Pick".
- Pick's_theorem label "Wzór Picka".
- Pick's_theorem label "Формула Пика".
- Pick's_theorem label "ピックの定理".
- Pick's_theorem label "皮克定理".
- Pick's_theorem sameAs Pickův_vzorec.
- Pick's_theorem sameAs Satz_von_Pick.
- Pick's_theorem sameAs Teorema_de_Pick.
- Pick's_theorem sameAs Théorème_de_Pick.
- Pick's_theorem sameAs Teorema_di_Pick.
- Pick's_theorem sameAs ピックの定理.
- Pick's_theorem sameAs Formule_van_Pick.
- Pick's_theorem sameAs Wzór_Picka.
- Pick's_theorem sameAs Teorema_de_Pick.
- Pick's_theorem sameAs m.01wg2w.
- Pick's_theorem sameAs Q646523.
- Pick's_theorem sameAs Q646523.
- Pick's_theorem sameAs Pick's_theorem.
- Pick's_theorem wasDerivedFrom Pick's_theorem?oldid=588037365.
- Pick's_theorem depiction Pick-theorem.png.
- Pick's_theorem isPrimaryTopicOf Pick's_theorem.
