(a) Triangles ABC and ABD are equilateral triangles, so have internal angles of 60°. The angle CBD is the sum of the measures of angles CBA and ABD, both of which are 60°.
angle CBD measures 120° = 2π/3 radians
(b) The area of the left shaded area is the area of circle A minus twice the area of circular segment CBD. The area of a circular segment that subtends an arc of α radians is
... A = (1/2)r²(α - sin(α))
Then the area of the left shaded area is
... (area of circle) - 2 × (area of segment)
... = π·r² - r²(2π/3 - sin(2π/3)) = r²(π/3 + sin(2π/3))
For a radius of 6 cm, the area of the left shaded area is
... (6 cm)²(π/3 + (√3)/2) ≈ 68.876 cm²
Then the area of both shaded areas is
... shaded area ≈ 2 × 68.876 cm² ≈ 137.752 cm²
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(If you erroneously use the 3-digit value 3.14 for π, then you will get the erroneous 4-digit number 137.7 cm² for the shaded area. The number of significant digits in your value of π should be at least the number of significant digits you want in your answer. For the correct 4-digit answer 137.8 cm², you should use at least a 4-digit value for π, such as 3.142.)
