Right Trapezoid

Author

Eric W. Weisstein
July 31, 2014

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/RightTrapezoid.nb.

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

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

Definitions

ineq=y≥0&&a≥x&&a h1+h2 x≥h1 x+a y&&x≥0;

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

verts={{0,0},{a,0},{a,h2},{0,h1}};

reg=Polygon[verts]

reg2=RegionDifference[Rectangle[{0,0},{a,h2}],Triangle[{{0,h1},{a,h2},{0,h2}}]];

assum=a>0&&h1>0&&h2>0;

Plots

Diagram

LaminaData["FilledRightTrapezoid","Diagram"]

DiscretizeRegion

Block[{a=1,h1=.3,h2=.6},DiscretizeRegion[#]]&/@{reg,reg2,implreg}

Schematic Figure

trap=Line[v={
{0,0},{a,0},{a,h2},{0,h1},
{0,0}
}];

Block[{a=1,h1=.3,h2=.6},
Graphics[{
trap,
{Dashing[{.03,.015}],Line[{{0,h1},{0,h2},{a,h2}}]},
Text[Style["a",Italic],{a/2,-.07}],
Text[Style[Subscript[Style["h",Italic],1]],{-.07,h1/2}],
Text[Style[Subscript[Style["h",Italic],2]],{a+.07,h2/2}],
Text[Style[Subscript[Style["h",Italic],2]-Subscript[Style["h",Italic],1]],{-.17,h1+(h2-h1)/2}],
PerpSquare[{Line[v[[{4,1}]]],Line[v[[{1,2}]]]},.07,4],
PerpSquare[{Line[v[[{1,2}]]],Line[v[[{2,3}]]]},.07,4]
}]]

Polygon

p=Polygon[{{0,0},{a,0},{a,h2},{0,h1}}];

Block[{a=1,h1=.3,h2=.6},Graphics[p]]

RegionPlot

Block[{a=1,h1=.3,h2=.6},RegionPlot[ineq,{x,0,a},{y,0,Max[h1,h2]},AspectRatio->Automatic]]

Equations

Inequality

FullSimplify[Inequalities[p,{x,y}],a>0&&h1>0&&h2>0]

Properties

Area

Area

Assuming[assum,FullSimplify[Area[reg]]]//Timing

Assuming[assum,FullSimplify[Area[implreg]]]//Timing

Integrate

Integrate[Boole[ineq[x,y]],{x,-∞,∞},{y,-∞,∞},GenerateConditions->True,Assumptions->a>0&&h1>0&&h2>0]//Simplify

RegionMeasure

Assuming[assum,FullSimplify[RegionMeasure[reg]]]

Assuming[assum,FullSimplify[RegionMeasure[implreg]]]

AreaInertiaTensor

V8

Integrate[AreaInertiaTensor[{x,y}]Boole[ineq[x,y]],{x,-∞,∞},{y,-∞,∞},GenerateConditions->True,Assumptions->a>0&&h1>0&&h2>0]

FullSimplify[%,a>0&&h1>0&&h2>0]

Centroid

Integrate

(1/((1/2) a (h1+h2)))Integrate[{x,y}Boole[ineq[x,y]],{x,-∞,∞},{y,-∞,∞},GenerateConditions->True,Assumptions->a>0&&h1>0&&h2>0]//FullSimplify

RegionCentroid

Assuming[assum,FullSimplify[RegionCentroid[reg]]]//Timing

Assuming[assum,FullSimplify[RegionCentroid[implreg]]]//Timing

Convex

TimeConstrained[Region`ConvexRegionQ[reg],600]//Timing

TimeConstrained[Region`ConvexRegionQ[implreg],600]//Timing

Diagonals

FullSimplify[Norm/@{{a,h2},{a,0}-{0,h1}},a>0&&h1>0&&h2>0]

EdgeLengths

s=FullSimplify[Norm/@(Subtract@@@Partition[verts,2,1,1]),assum]

GeneralizedDiameter

FullSimplify[Max[a,(Sqrt[(a^2)+(h2^2)]),(Sqrt[(a^2)+(h1^2)])],a>0&&h1>0&&h2>0]

Perimeter

Total[s]

RadiiOfGyration

Assuming[a>0&&h1>0&&h2>0,FullSimplify[RadiiOfGyration[ineq[x,y],{x,y}]]]//Timing