The semicircle shown at left has center X XX and diameter W Z ‾ ​WZ ​ ​​ start overline, W, Z, end overline. The radius X Y ‾ ​XY ​ ​​ start overline, X, Y, end overline of the semicircle has length 2 22. The chord Y Z ‾ ​YZ ​ ​​ start overline, Y, Z, end overline has length 2 22. What is the area of the shaded sector formed by obtuse angle W X Y WXYW, X, Y?

Triangle XYZ is equilateral, so the central angle of the sector is 120°, or 2π/3 radians. The area of a sector of central angle β is given by
.. A = (1/2)r^2*β . . . . . . β in radians
Your sector's area is
.. A = (1/2)*2^2*(2π/3) = 4π/3 . . . . square units