M2 Trigonometry

area of ΔABC = (1/2)(AB)(BC)(sin∠ABC)
= (1/2)(2a)²(sin60')
= (√3)*a²

note that ΔAOD and ΔBOE are equilateral triangles,
and also ΔAOD and ΔBOE are congruent triangles

area of ΔAOD = area of ΔBOE
= (1/2)(a²)(sin60')
= (√3)*a² /4
(since OA=OB and AB = 2a, so OA =a)

note that ∠DOE = 60' as ∠DOA =∠BOE =60'
area of sector DOE = πa² * (∠DOE /360')
= πa² /6

area of shaded region
= area of ΔABC - area of ΔAOD - area of ΔBOE - area of sector DOE
= (√3)*a² - (√3)*a²/4 - (√3)*a²/4 - πa²/6
= (√3)*a²/2 - πa²/6
= [(3√3) - π]*a² /6

(√3/4)(2a)²- (√3/4)(a)²- (1/6) π(a)²-(√3/4)(a)²=(√3/2-π/6)a²