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Four equal circles are described about the four corners of a square so that each touches two of the others, as shown in the figure. Find the area of the shaded region, if each side of the square measures 14 cm.
Here the distance between the center of circles touching each other is equal to the side of the square. Therefore, we can say that the radius of ach circle is equal to the half of the side of the square. Now by simply calculating the area of the 4 quadrants and then subtracting it from the area of the square we can easily calculate the area of the shaded region.
Given side of square = a = 14 cm
Central angle of each sector formed at corner = θ = 90°
So, radius of 4 equal circles = r = a/2 = 14/2
∴ Radius of 4 circles = r = 7 cm
where R = radius of circle
⇒ Area of all 4 quadrants = 49π → eqn2
Also, Area of square = side×side = a×a = a2 = 142 (putting value of side of square)
⇒ Area of square = 196 cm2→ eqn3
∴ Area of shaded region = Area of square – Area of all 4 quadrants
⇒ Area of shaded region = 196 – 49π (fromeqn3 and eqn2)
= 196 – (7×22)
= 196 – 154
= 42 cm2
The area of shaded region is 42 cm2.
ABCD is a field in the shape of a trapezium, AD || BC, ∠ABC = 90° and ∠ADC = 60°. Four sectors are formed with centres A, B, C and D, as shown in the figure. The radius of each sector is 14 m.
Find the following:
(i) total area of the four sectors,
(ii) area of the remaining portion, given that AD = 55 m, BC = 45 m and AB = 30 m.
A child draws the figure of an aeroplane as shown. Here, the wings ABCD and FGHI are parallelograms, the tail DEF is an isosceles triangle, the cockpit CKI is a semicircle and CDFI is a square. In the given figure, BP ⊥ CD, HQ ⊥ FI and EL ⊥ DF. If CD = 8 cm, BP = HQ = 4 cm and DE = EF = 5 cm, find the area of the whole figure. [Take π = 3.14.]