Can I hatch this region in any way?

Update: Version 12.1 comes with new directives PatternFilling and HatchFilling that can be use as the option setting for PlotStyle:

RegionPlot[reg, Mesh -> None, 
  PlotStyle -> PatternFilling["Diamond", ImageScaled[.025]], 
  BoundaryStyle -> Directive[Thick, Red], 
  Prolog -> {Thick, Circle[{0, 0}, 10]},
  PlotRangePadding -> Scaled[.05], PlotRange -> {-10, 10}]

RegionPlot[reg, Mesh -> None, 
 PlotStyle -> Directive[Gray, HatchFilling["Diagonal", 3, 5]], 
 BoundaryStyle -> Directive[Thick, Red], 
 Prolog -> {Thick, Circle[{0, 0}, 10]}, 
 PlotRangePadding -> Scaled[.05], PlotRange -> {-10, 10}, 
 PlotPoints -> 100]

Original answer:

reg = ImplicitRegion[x^2 + y^2 <= 100 && 5 <= x <= 5 Sqrt[3], {x, y}];
N[Area[reg]]

52.3599

Show[Graphics[{Gray, Circle[{0, 0}, 10]}, Axes -> True], 
 RegionPlot[reg, MeshFunctions -> {# + #2 &, # - #2 &}, 
  Mesh -> {50, 50}, MeshShading -> None, PlotStyle -> None, 
  BoundaryStyle -> Red]]

RegionPlot[x^2 + y^2 <= 100 && 5 <= x <= 5 Sqrt[3],{x, -10, 10}, {y, -10, 10}, 
 MeshFunctions -> {# + #2 &, # - #2 &}, Mesh -> {50, 50}, 
 MeshShading -> None, PlotStyle -> None, BoundaryStyle -> Red, 
 PlotPoints -> 90, Axes -> True, Epilog -> {Gray, Scale[Circle[], 10]}, 
 Frame -> False]

same picture

poly = MeshPrimitives[
    BoundaryDiscretizeRegion[
     RegionIntersection[
      Disk[],
      HalfPlane[{{5/10, 0}, {5/10, 1}}, {1, 0}],
      HalfPlane[{{5 Sqrt[3]/10, 0}, {5 Sqrt[3]/10, 1}}, {-1, 0}]
      ],
     MaxCellMeasure -> {1 -> 0.001}
     ],
    2
    ][[1]];
Area[poly]

0.523599

Graphics[{Circle[], Gray, EdgeForm[{Thick, Black}], poly}]

   Graphics[{Red, [email protected], Disk[{0, 0}, 10], Opacity@1, Blue, Thick, 
      Circle[{0, 0}, 10, {π/6, π/3}], 
      Circle[{0, 0}, 10, {-(π/6), -(π/3)}], Green, [email protected], 
      Rectangle[{10 Cos[π/3], -10 Sin[π/3]}, {10 Cos[π/6], 
        10 Sin[π/3]}]}, Axes -> True, AxesOrigin -> {0, 0}]

Therefore we can use the following.

reg1 = Disk[{0, 0}, 10];
reg2 = Rectangle[{10 Cos[π/3], -10 Sin[π/3]}, {10 Cos[π/ 6], 10 Sin[π/3]}];
reg=RegionIntersection[reg1, reg2];  
 Area@reg

$\frac{50 \pi }{3}$

Show[Graphics[{Circle[{0, 0}, 10]}, Axes -> True], 
 Region[reg, BaseStyle -> {LightBlue, EdgeForm[{Red, Thick}]}]]

You can also choose reg2 as

reg2 = Rectangle[{5, -5 Sqrt[3]}, {5 Sqrt[3], 5 Sqrt[3]}];

Or

reg2 = Rectangle[{5, -10}, {5 Sqrt[3], 10}];