Half-Disk

Author

Eric W. Weisstein
August 29, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/Laminae/Half-Disk.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/Half-Disk.html.

©2026 Wolfram Research, Inc. except for portions noted otherwise

Definitions

ineq=x^2+y^2<=a^2&&y≥0;

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

reg=Disk[{0,0},a,{0,Pi}];

reg2=RegionIntersection[ImplicitRegion[y≥0,{x,y}],Disk[{0,0},a]];

assum=a>0;

Plots

Diagram

Graphics[{{FaceForm[LightYellow],EdgeForm[None],Disk[{0,0},1,{0,π}]},{LightDarkSwitched[Black,White],AbsoluteThickness[2],Line[{{-1,0},{1,0}}],Circle[{0,0},1,{0,π}]}}]

Properties

Area

Area

Assuming[assum,Area[reg]]

Assuming[assum,Area[reg2]]

Assuming[assum,Area[implreg]]

Assuming[assum,Area[EntityValue[Entity["Lamina","HalfDisk"],"ImplicitRegion"][a]]]

Assuming[assum,Area[EntityValue[Entity["Lamina","HalfDisk"],"ParametricRegion"][a]]]

Assuming[assum,Area[EntityValue[Entity["Lamina","HalfDisk"],"Region"][a]]]

Integrate

area=Assuming[a>0,Area[ineq,{x,y}]]

RegionMeasure

Assuming[assum,RegionMeasure[reg]]

Assuming[assum,RegionMeasure[reg2]]

Assuming[assum,RegionMeasure[implreg]]

AreaInertiaTensor

Assuming[a>0,AreaInertiaTensor[ineq,{x,y}]]

Centroid

Integrate

Assuming[a>0,Centroid[ineq,{x,y}]]

RegionCentroid

Assuming[assum,RegionCentroid[reg]]

Assuming[assum,RegionCentroid[reg2]]

Assuming[assum,RegionCentroid[implreg]]

Convex

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

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

GeneralizedDiameter

ineqs=And@@ineq@@@{{x1,y1},{x2,y2}}/.a->1

Maximize[{(x1-x2)^2+(y1-y2)^2,ineqs},{x1,x2,y1,y2}]//Timing

RadiiOfGyration

Assuming[a>0,FullSimplify[RadiiOfGyration[ineq[x,y],{x,y}]]]

Regions

EntityValue[Entity["Lamina","HalfDisk"],"ImplicitRegion"][a]

EntityValue[Entity["Lamina","HalfDisk"],"ParametricRegion"][a]

EntityValue[Entity["Lamina","HalfDisk"],"Region"][a]