Isosceles Trapezoid

Author

Eric W. Weisstein
July 23, 2014

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polygons/IsoscelesTrapezoid.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/IsoscelesTrapezoid.html.

©2014 Wolfram Research, Inc. except for portions noted otherwise

Code

IsoscelesTrapezoid[{a_,b_}]:=IsoscelesTrapezoid[{a,b},1]
IsoscelesTrapezoid[{a_,b_},h_]:=Quadrilateral[{{-a/2,0},{-b/2,h},{b/2,h},{a/2,0}}]

Definitions

ineq=y≥0&&b (Sqrt[-(a-b)^2+4 c^2]-2 y)+2 a y≥2 Sqrt[-(a-b)^2+4 c^2] x&&a Sqrt[-(a-b)^2+4 c^2]≥2 a y&&Sqrt[-(a-b)^2+4 c^2] (b+2 x)+2 a y≥2 b y;

implreg=ImplicitRegion[ineq,{x,y}];

reg=With[{h=Sqrt[c^2-(b-a)^2/4]},Polygon[{{-b/2,0},{b/2,0},{a/2,h},{-a/2,h}}]];

assum=0<a<b&&c>0;

Figure

trap=With[{h=Sqrt[c^2-(b-a)^2/4]},Line[v={{-b/2,0},{b/2,0},{a/2,h},{-a/2,h},{-b/2,0}}]]

Block[{a=1,b=1.5,c=.75},Graphics[{
trap,
EqualAngleTicks[v[[#]],1,.15,.04,1]&/@{{4,1,2},{1,2,3}},
EqualSideTicks[v[[{1,4}]],1,.04,0],
EqualSideTicks[v[[{2,3}]],1,.04,.04],
Text[Style["b",Italic],{0,-.1}],
Text[Style["a",Italic],{0,.1+(h=(Sqrt[-(1/4) (((-a+b))^2)+(c^2)]))}],
Text[Style["c",Italic],{.75#,h/2}]&/@{-1,1}

}]]

Plots

Diagram

Show[LaminaData["FilledIsoscelesTrapezoid","Diagram"],Axes->False]

DiscretizeRegion

Block[{a=1,b=1.5,c=.7},Show[DiscretizeRegion[reg],ImageSize->Small]]

Block[{a=1,b=1.5,c=.7},Show[DiscretizeRegion[implreg],ImageSize->Small]]

Polygon

p=Polygon[{{-b/2,0},{b/2,0},{a/2,(Sqrt[(c^2)-(1/4)(((b-a))^2)])},{-a/2,(Sqrt[(c^2)-(1/4)(((b-a))^2)])}}];

Block[{a=1,b=1.5,c=.75},Graphics[p]]

FullSimplify[Inequalities[p,{x,y}],b>0&&Sign[-(1/4) (((a-b))^2)+(c^2)]==1]

RegionPlot

ineq[x_,y_]:=y≥0&&b ((Sqrt[-(((a-b))^2)+4 (c^2)])-2 y)+2 a y≥2 (Sqrt[-(((a-b))^2)+4 (c^2)]) x&&a (Sqrt[-(((a-b))^2)+4 (c^2)])≥2 a y&&(Sqrt[-(((a-b))^2)+4 (c^2)]) (b+2 x)+2 a y≥2 b y

Block[{a=1,b=1.5,c=.75},RegionPlot[ineq[x,y],{x,-b/2,b/2},{y,0,(Sqrt[(c^2)-(1/4)(((b-a))^2)])},AspectRatio->Automatic]]

Properties

Area

Area

Assuming[assum,Simplify@Area[reg]]

Integrate

Integrate[Boole[ineq[x,y]],{x,-∞,∞},{y,-∞,∞},GenerateConditions->True,Assumptions->0<a<b&&c>0&&(c^2)-(1/4)(((b-a))^2)>0]

area=FullSimplify[%,0<a<b&&c>0&&(c^2)-(1/4)(((b-a))^2)>0]

RegionMeasure

Assuming[assum,Simplify@RegionMeasure[reg]]

Diagonals

FullSimplify[Norm[{-b/2,0}-{a/2,(Sqrt[(c^2)-(1/4)(((b-a))^2)])}],a>0&&b>0&&c>0&&0<a<b&&(c^2)-(1/4)(((b-a))^2)>0]

GeneralizedDiameter

FullSimplify[Max[b,(Sqrt[a b+(c^2)])],a>0&&b>0&&b>a&&c>0]

Centroid

(1/((1/4) (a+b) (Sqrt[-(((a-b))^2)+4 (c^2)])))Integrate[{x,y}Boole[ineq[x,y]],{x,-∞,∞},{y,-∞,∞},GenerateConditions->True,Assumptions->0<a<b&&c>0&&(c^2)-(1/4)(((b-a))^2)>0]

FullSimplify[%,0<a<b&&c>0&&(c^2)-(1/4)(((b-a))^2)>0]

InertiaTensor

V7

Integrate[AreaInertiaTensor[{x,y}]Boole[ineq[x,y]],{x,-∞,∞},{y,-∞,∞},GenerateConditions->True,Assumptions->0<a<b&&c>0&&(c^2)-(1/4)(((b-a))^2)>0]

FullSimplify[%,0<a<b&&c>0&&(c^2)-(1/4)(((b-a))^2)>0]

V8

Integrate[AreaInertiaTensor[{x,y}]Boole[ineq[x,y]],{x,-∞,∞},{y,-∞,∞},GenerateConditions->True,Assumptions->0<a<b&&c>0&&(c^2)-(1/4)(((b-a))^2)>0]

FullSimplify[%,0<a<b&&c>0&&(c^2)-(1/4)(((b-a))^2)>0]

RadiiOfGyration

Assuming[0<a<b&&c>0&&(c^2)-(1/4)(((b-a))^2)>0,FullSimplify[RadiiOfGyration[ineq[x,y],{x,y}]]]//Timing

(((2(Sqrt[6])))^2)