Quarter-Disk

Author

Eric W. Weisstein
August 29, 2026

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

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

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

Definitions

ineq=x^2+y^2<=a^2&&y≥0&&x≥0;

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

reg=Disk[{0,0},a,{0,π/2}];

assum=a>0;

Plots

Diagram

Graphics[{{FaceForm[LightYellow],EdgeForm[None],Disk[{0,0},1,{0,(π/2)}]},{LightDarkSwitched[Black,White],AbsoluteThickness[2],Line[{{0,1},{0,0},{1,0}}],Circle[{0,0},1,{0,(π/2)}]}},ImageSize->250]

Properties

Area

Area

Assuming[assum,Area[reg]]

Integrate

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

RegionMeasure

Assuming[assum,RegionMeasure[reg]]

Centroid

Integrate

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

RegionCentroid

Assuming[assum,RegionCentroid[reg]]

InertiaTensor

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

RadiiOfGyration

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

GeneralizedDiameter

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

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

Lengths

(π/4)NIntegrate[(Sqrt[(x1-x2)^2+(((y1-y2))^2)])Boole[x1^2+y1^2<1&&x2^2+y2^2<1&&x1>0&&x2>0&&y1>0&&y2>0],{x1,0,1},{x2,0,1},{y1,0,1},{y2,0,1}]

(π/4)NIntegrate[(Sqrt[(x1-x2)^2+(((y1-y2))^2)])Boole[x1^2+y1^2<1&&x2^2+y2^2<1&&x1>0&&x2>0&&y1>0&&y2>0],{x1,0,1},{x2,0,1},{y1,0,1},{y2,0,1},Method->"QuasiMonteCarlo"]

V7

Integrate[(Sqrt[(x1-x2)^2+(((y1-y2))^2)])Boole[x1^2+y1^2<1&&x2^2+y2^2<1&&x1>0&&x2>0&&y1>0&&y2>0],{x1,0,1},Assumptions->0<x2<1&&0<y1<1&&0<y2<1]

i1=FullSimplify[%1,x1^2+y1^2<1&&x2^2+y2^2<1&&1>x1>0&&1>x2>0&&1>y1>0&&1>y2>0&&y1≠y2&&x1≠x2]

V8

Integrate[(Sqrt[(x1-x2)^2+(((y1-y2))^2)])Boole[x1^2+y1^2<1&&x2^2+y2^2<1&&x1>0&&x2>0&&y1>0&&y2>0],{x1,0,1},Assumptions->0<x2<1&&0<y1<1&&0<y2<1]

i1=FullSimplify[%4,x1^2+y1^2<1&&x2^2+y2^2<1&&1>x1>0&&1>x2>0&&1>y1>0&&1>y2>0&&y1≠y2&&x1≠x2]

i1=%;

(π/4)NIntegrate[i1 Boole[x2^2+y2^2<1&&x2>0&&y1>0&&y2>0],{x2,0,1},{y1,0,1},{y2,0,1},Method->"QuasiMonteCarlo"]

Integrate[(1/2) ((Sqrt[(-1+(y1^2)) (-1-(x2^2)+2 x2 (Sqrt[1-(y1^2)])+2 y1 y2-(y2^2))])+x2 ((Sqrt[(x2^2)+(((y1-y2))^2)])-(Sqrt[1+(x2^2)-2 x2 (Sqrt[1-(y1^2)])-2 y1 y2+(y2^2)]))-(((y1-y2))^2) (Log[-x2+(Sqrt[(x2^2)+(((y1-y2))^2)])]-Log[-x2+(Sqrt[1-(y1^2)])+(Sqrt[1+(x2^2)-2 x2 (Sqrt[1-(y1^2)])-2 y1 y2+(y2^2)])])),{y1,0,1},Assumptions->0<x2<1&&0<y2<1&&x2^2+y2^2<1]

Integrate[(1/2) ((Sqrt[(-1+(y1^2)) (-1-(x2^2)+2 x2 (Sqrt[1-(y1^2)])+2 y1 y2-(y2^2))])+x2 ((Sqrt[(x2^2)+(((y1-y2))^2)])-(Sqrt[1+(x2^2)-2 x2 (Sqrt[1-(y1^2)])-2 y1 y2+(y2^2)]))-(((y1-y2))^2) (Log[-x2+(Sqrt[(x2^2)+(((y1-y2))^2)])]-Log[-x2+(Sqrt[1-(y1^2)])+(Sqrt[1+(x2^2)-2 x2 (Sqrt[1-(y1^2)])-2 y1 y2+(y2^2)])])),y1]