Graphics – Can I hatch this region somehow?

            graphic[{Red, Opacity@0.7, 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, opacity@0.6,
rectangle[{10 Cos[π/3]-10 sin[π/3]}, {10 Cos[π/6],
10 Sin[π/3]}]}, Axis -> True, AxesOrigin -> {0, 0}]

Enter the image description here

That's why we can use the following.

reg1 = hard disk[{0, 0}, 10];
reg2 = rectangle[{10 Cos[π/3]-10 sin[π/3]}, {10 Cos[π/ 6]10 Sin[π/3]}];

Area @ RegionIntersection[reg1, reg2]

$ frac {50 pi} {3} $

You can also choose reg2 as

reg2 = rectangle[{5, -5 Sqrt[3]}, {5 Sqrt[3], 5 square meters[3]}];

Or

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