Regular Octagon
Author
Eric W. Weisstein
Title
Regular Octagon
Description
The regular octagon is the regular polygon with eight sides, as illustrated above. The inradius r, circumradius R, and area A of the regular octagon can be computed directly from the formulas for a general regular polygon with side length a and n=8 sides as r = 1/2acot(pi/8) (1) = 1/2(1+sqrt(2))a (2) R = 1/2acsc(pi/8) (3) = 1/2sqrt(4+2sqrt(2))a (4) A = 1/4na^2cot(pi/8) (5) = 2(1+sqrt(2))a^2. (6) The vertex angle alpha, central angle theta, and exterior angle beta are given by theta...
Category
Educational Materials
Keywords
URL
http://www.notebookarchive.org/2019-07-0z5jt4u/
DOI
https://notebookarchive.org/2019-07-0z5jt4u
Date Added
2019-07-02
Date Last Modified
2019-07-02
File Size
227.48 kilobytes
Supplements
Rights
Redistribution rights reserved
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Regular Octagon
Regular Octagon
Author
Author
Eric W. Weisstein
July 3, 2018
July 3, 2018
©2018 Wolfram Research, Inc. except for portions noted otherwise
Definitions
Definitions
ineq=a+(-2+Sqrt[2])(x+y)≥0&&a+Sqrt[2]a≥2y&&-2(-1+Sqrt[2])(x-y)≤Sqrt[2]a&&a+Sqrt[2]a+2x≥0&&-2(-1+Sqrt[2])(x+y)≤Sqrt[2]a&&a+Sqrt[2]a+2y≥0&&a+(-2+Sqrt[2])(x-y)≥0&&a+Sqrt[2]a≥2x;
implreg=ImplicitRegion[ineq,{x,y}];
verts=a{{1/2Cot[Pi/8],1/2},{1/2,1/2Cot[Pi/8]},{-(1/2),1/2Cot[Pi/8]},{-(1/2)Cot[Pi/8],1/2},{-(1/2)Cot[Pi/8],-(1/2)},{-(1/2),-(1/2)Cot[Pi/8]},{1/2,-(1/2)Cot[Pi/8]},{1/2Cot[Pi/8],-(1/2)}};
reg=Polygon[verts];
assum=a>0;
Figure
Figure
In[]:=
Show[LaminaData["FilledRegularOctagon","Diagram"],ImageSize300]
Out[]=
Plots
Plots
Properties
Properties
TableForm[RegularPolygonInformation[8]//Simplify,TableDepth2]
sides n | 8 |
vertex angle α {rad, degrees} | 3π 4 |
central angle β {rad, degrees} | π 4 |
inradius r | 1 2 π 8 |
circumradius R | 1 2 π 8 |
area A | 2Cot π 8 |
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General::compat:Combinatorica Graph and Permutations functionality has been superseded by preloaded functionaliy.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
0.027227,2(1+
2
)Assuming[assum,FullSimplify[Area[implreg/.a1]]]//Timing
0.035630,2(1+
2
)Integrate
Integrate
Assuming[a>0,Area[ineq[x,y],{x,y}]]//FullSimplify//Timing
1.0275,2(1+
2
)2
a
RegionMeasure
RegionMeasure
Assuming[assum,FullSimplify[RegionMeasure[reg]]]
RegionMeasure::nmet:Unable to compute the measure of region PolygonaCot,,,aCot,-,aCot,-aCot,,-aCot,-,-,-aCot,,-aCot,aCot,-.
1
2
π
8
a
2
a
2
1
2
π
8
a
2
1
2
π
8
1
2
π
8
a
2
1
2
π
8
a
2
a
2
1
2
π
8
a
2
1
2
π
8
1
2
π
8
a
2
RegionMeasurePolygonaCot,,,aCot,-,aCot,-aCot,,-aCot,-,-,-aCot,,-aCot,aCot,-
1
2
π
8
a
2
a
2
1
2
π
8
a
2
1
2
π
8
1
2
π
8
a
2
1
2
π
8
a
2
a
2
1
2
π
8
a
2
1
2
π
8
1
2
π
8
a
2
Assuming[assum,FullSimplify[RegionMeasure[implreg]]]
2(1+
2
)2
a
AreaInertiaTensor
AreaInertiaTensor
Assuming[a>0,AreaInertiaTensor[ineq[x,y],{x,y}]]//FullSimplify//Timing
6.17734,(11+8,0,0,(11+8
1
12
2
)4
a
1
12
2
)4
a
FullSimplifyn2+CosCot,0,0,nCot1+3/.n8
1
192
4
a
2π
n
π
n
2
Csc
π
n
1
192
4
a
π
n
2
Cot
π
n
(11+8,0,0,(11+8
1
12
2
)4
a
1
12
2
)4
a
Centroid
Centroid
Integrate
Integrate
Assuming[a>0,Centroid[ineq[x,y],{x,y}]]//FullSimplify//Timing
{5.43524,{0,0}}
RegionCentroid
RegionCentroid
Assuming[assum,FullSimplify[RegionCentroid[reg/.a1]]]//Timing
{0.005529,{0,0}}
Assuming[assum,FullSimplify[RegionCentroid[implreg/.a1]]]//Timing
{0.041418,{0,0}}
Convex
Convex
TimeConstrained[Region`ConvexRegionQ[reg],600]//Timing
TimeConstrained[Region`ConvexRegionQ[implreg],600]//Timing
Diagonals
Diagonals
EdgeLengths
EdgeLengths
s=FullSimplify[Norm/@(Subtract@@@Partition[verts,2,1,1]),assum]
{a,a,a,a,a,a,a,a}
GeneralizedDiameter
GeneralizedDiameter
GD[n_]:=IfOddQ[n],Csc,Csca
1
2
π
2n
π
n
GD[8]//FunctionExpand//FullSimplify
2(2+
a2
)ineqs=FullSimplify[And@@ineq@@@{{x1,y1},{x2,y2}},a>0]/.a1
1+(-2+
2
)(x1+y1)≥0&&1+2
≥2y1&&-2(-1+2
)(x1-y1)≤2
&&1+2
+2x1≥0&&-2(-1+2
)(x1+y1)≤2
&&1+2
+2y1≥0&&1+(-2+2
)(x1-y1)≥0&&1+2
≥2x1&&1+(-2+2
)(x2+y2)≥0&&1+2
≥2y2&&-2(-1+2
)(x2-y2)≤2
&&1+2
+2x2≥0&&-2(-1+2
)(x2+y2)≤2
&&1+2
+2y2≥0&&1+(-2+2
)(x2-y2)≥0&&1+2
≥2x2Maximize[{(x1-x2)^2+(y1-y2)^2,ineqs},{x1,x2,y1,y2},Reals]//Timing
$Aborted
Perimeter
Perimeter
Total[s]
8a
RadiiOfGyration
RadiiOfGyration
Cite this as: Eric W. Weisstein, "Regular Octagon" from the Notebook Archive (2018), https://notebookarchive.org/2019-07-0z5jt4u
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