Regular Hexagon
Author
Eric W. Weisstein
Title
Regular Hexagon
Description
The regular hexagon is the regular polygon with six sides, as illustrated above. The inradius r, circumradius R, sagitta s, and area A of a regular hexagon can be computed directly from the formulas for a general regular polygon with side length a and n=6 sides, r = 1/2acot(pi/6) (1) = 1/2sqrt(3)a (2) R = 1/2acsc(pi/6) (3) = a (4) s = 1/2atan(pi/(12)) (5) = 1/2(2-sqrt(3))a (6) A = 1/4na^2cot(pi/6) (7) = 3/2sqrt(3)a^2. (8) Therefore, for a regular hexagon, ...
Category
Educational Materials
Keywords
URL
http://www.notebookarchive.org/2019-07-0z5js14/
DOI
https://notebookarchive.org/2019-07-0z5js14
Date Added
2019-07-02
Date Last Modified
2019-07-02
File Size
131.24 kilobytes
Supplements
Rights
Redistribution rights reserved
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Regular Hexagon
Regular Hexagon
Author
Author
Eric W. Weisstein
July 3, 2018
July 3, 2018
©2018 Wolfram Research, Inc. except for portions noted otherwise
Code
Code
Hexagon::usage="Hexagon[{a, b, c}, {t1, t2}] gives a hexagon with side lengths {a,b,c,a,b,c} and opening angles {t1,-t2}."HexagonTriangles::usage="HexagonTriangles[{v1,...,v6}] gives the two congruent triangles obtained by taking each three sides and drawing a fourth point to make a parallelogram."
Hexagon[{a_,b_,c_},{t1_,t2_}]:=Module[{aa=a{Cos[t1],Sin[t1]},bb=b{Cos[t2],-Sin[t2]},cc=c{0,-1}},Line[FoldList[Plus,{0,0},{aa,bb,cc,-aa,-bb,-cc}]]]HexagonTriangles[a_List/;Length[a]6]:=Module[{l=Join[a,Take[a,3]],i,p},p=Table[l[[i+2]]+l[[i]]-l[[i+1]],{i,6}];{{PointSize[.03],Point/@p},ClosedLine[p[[{1,3,5}]]],Triangle[p[[{2,4,6}]]]}]
Definitions
Definitions
ineq=Sqrt[3]a≥Sqrt[3]x+y&&Sqrt[3]a≥2y&&Sqrt[3](a+x)≥y&&Sqrt[3]a+Sqrt[3]x+y≥0&&Sqrt[3]a+2y≥0&&Sqrt[3]a+y≥Sqrt[3]x;
implreg=ImplicitRegion[ineq,{x,y}];
verts=a{{1,0},{1/2,Sqrt[3]/2},{-(1/2),Sqrt[3]/2},{-1,0},{-(1/2),-(Sqrt[3]/2)},{1/2,-(Sqrt[3]/2)}};
reg=Polygon[verts];
assum=a>0;
Figure
Figure
In[]:=
Show[LaminaData["FilledRegularHexagon","Diagram"],ImageSize300]
Out[]=
Diagrams
Diagrams
Diagram
Diagram
LaminaData["FilledRegularHexagon","Diagram"]
DiscretizeRegion
DiscretizeRegion
Block[{a=1},DiscretizeRegion[#]]&/@{reg,implreg}
,
RegionPlot
RegionPlot
RegionPlot[ineq/.a1,{x,-1,1},{y,-1,1}]
Polygon
Polygon
Graphics[RegularPolygon[6]]
InequalityPlot
InequalityPlot
<<Graphics`InequalityGraphics`<<Utilities`Typesetting`
InequalityPlot[#<1.,{x},{y},TicksNone,PlotLabelTraditionalForm[#<1]]&[2Abs[x]+Abs[x-
3
y]+Abs[x+3
y]]⁃Graphics⁃
Properties
Properties
TableForm[RegularPolygonInformation[6]//Simplify,TableDepth2]//TraditionalForm
sides n | 6 |
vertex angle α {rad, degrees} | 2π 3 |
central angle β {rad, degrees} | π 3 |
inradius r | 3 2 |
circumradius R | 1 |
sagitta s | 1- 3 2 |
area A | 3 3 2 |
Equations
Equations
Properties
Properties
General::compat:Combinatorica Graph and Permutations functionality has been superseded by preloaded functionaliy. The package now being loaded may conflict with this. Please see the Compatibility Guide for details.
Area
Area
Area
Area
Assuming[assum,FullSimplify[Area[reg/.a1]]]//Timing
0.012839,
3
3
2
Assuming[assum,FullSimplify[Area[implreg/.a1]]]//Timing
0.019708,
3
3
2
Integrate
Integrate
Assuming[a>0,Area[ineq,{x,y}]]//FullSimplify//Timing
0.41732,
3
3
2
a
2
RegionMeasure
RegionMeasure
Assuming[assum,FullSimplify[RegionMeasure[reg]]]
RegionMeasure::nmet:Unable to compute the measure of region Polygon{a,0},,,-,,{-a,0},-,-,,-.
a
2
3
a2
a
2
3
a2
a
2
3
a2
a
2
3
a2
RegionMeasurePolygon{a,0},,,-,,{-a,0},-,-,,-
a
2
3
a2
a
2
3
a2
a
2
3
a2
a
2
3
a2
Assuming[assum,FullSimplify[RegionMeasure[reg/.a1]]]
3
3
2
Assuming[assum,FullSimplify[RegionMeasure[implreg]]]
3
3
2
a
2
AreaInertiaTensor
AreaInertiaTensor
Assuming[a>0,AreaInertiaTensor[ineq,{x,y}]]//FullSimplify//Timing
1.67496,,0,0,
5
3
4
a
16
5
3
4
a
16
Centroid
Centroid
Integrate
Integrate
Assuming[a>0,Centroid[ineq,{x,y}]]//FullSimplify//Timing
{1.64506,{0,0}}
RegionCentroid
RegionCentroid
Assuming[assum,FullSimplify[RegionCentroid[reg/.a1]]]//Timing
{0.003793,{0,0}}
Assuming[assum,FullSimplify[RegionCentroid[implreg/.a1]]]//Timing
{0.021668,{0,0}}
Diagonals
Diagonals
EdgeLengths
EdgeLengths
s=FullSimplify[Norm/@(Subtract@@@Partition[verts,2,1,1]),assum]
{a,a,a,a,a,a}
GeneralizeDiameter
GeneralizeDiameter
Perimeter
Perimeter
Total[s]
6a
RadiiOfGyration
RadiiOfGyration
Assuming[a>0,FullSimplify[RadiiOfGyration[ineq[x,y],{x,y}]]]//Timing
2.61021,a,a
1
2
5
6
1
2
5
6
Cite this as: Eric W. Weisstein, "Regular Hexagon" from the Notebook Archive (2018), https://notebookarchive.org/2019-07-0z5js14
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