Area of shaded region = area of circle - area of segment
(where "segment" refers to the unshaded region)
Area of circle = π (11.1 m)² ≈ 387.08 m²
The area of the segment is equal to the area of the sector that contains it, less the area of an isosceles triangle:
Area of segment = area of sector - area of triangle
130° is 13/36 of a full revolution of 360°. This is to say, the area of the sector with the central angle of 130° has a total area equal to 13/36 of the total area of the circle, so
Area of sector = 13/36 π (11.1 m)² ≈ 139.78 m²
Use the law of cosines to find the length of the chord (the unknown side of the triangle, call it x) :
x ² = (11.1 m)² + (11.1 m)² - 2 (11.1 m)² cos(130°)
x ² ≈ 404.82 m²
x = 20.12 m
Call this length the base of the triangle. Use a trigonometric relation to determine the corresponding altitude/height, call it y. With a vertex angle of 130°, the two congruent base angles of the triangle each measure (180° - 130°)/2 = 25°, so
sin(25°) = y / (11.1 m)
y = (11.1 m) sin(25°)
y ≈ 4.69 m
Then
Area of triangle = xy/2 ≈ 1/2 (20.12 m) (4.69 m) ≈ 47.19 m²
so that
Area of segment ≈ 139.78 m² - 47.19 m² ≈ 92.59 m²
Finally,
Area of shaded region ≈ 387.08 m² - 92.59 m² ≈ 294.49 m²
