find the area of the shaded region in the above mentioned picture
Answers
Answer:
60.28
60.3 (rounded off to one decimal)
Step-by-step explanation:
Let say semi circle touches rectangle at A & B point
C & D are the points where semi-circles intersect
Area above chord CD and Below Chord CD are same as Semicircles are identical
we will calculate the area above chord CD and multiply by 2 get the total shaded region area
we have a Area of region BACD
Area above chord CD = Area of region BACD - Area of Δ BCD
Area of region BACD = {(∠CBD)/360} × pie × Radius²
Lets calculate ∠CBD first
BC = BD = BA = Radius = 7
Let say AB & CD bisect at point O
BO = 7/2
so cos∠CBO = Base/hypoteneus = (7/2)/7 = 1/2
Cos 60° = 1/2 so ∠CBO = 60°
Similarly ∠DBO = 60°
∠CBD = 60 + 60 = 120°
Area of region BACD = (120/360) × pie × Radius²
= (1/3)(22/7)×7²
= 154/3
= 51.33 units Eq 1
in right angle Δ BOC
Sin 60° = Perpendicular / hypotenuse = CO/BC
√3 / 2 = CO / 7
CO = 7√3 / 2
Similarly DO = 7√3 /2
CD = 7√3 /2 + 7√3 /2 = 7√3
while BO = 7/2
Area of Δ BCD = (1/2) × Base × Height
= (1/2) (7√3) × (7/2)
= 49 √3/4
= 49 × 1.73 / 4
= 21.19 Eq 2
Area above chord CD = Area of region BACD - Area of Δ BCD
putting values from Eq 1 & Eq 2
Area above chord CD = 51.33 - 21.19 = 30.14
Total area of shaded region = 2 × 30.14 = 60.28
= 60.3 Units