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If a square is inscribed in a circle, find the ratio of the areas of the circle and the square.
Consider the figure shown below where O is centre of circle, join BC which passes through O, let the side of square be ‘a’ and radius of circle be ‘r’.
Now we know OB and OC are radius of circle
So, OB = OC = r
Consider ∆BDC right angled at D
And we know BC = OC + OB
BC = 2r and BD = DC = a (put these values in eqn1)
⇒ (2r)2 = a2 + a2
⇒ 4r2 = 2a2
Area of inscribed square = side × side
Areaa of inscribed square = a × a
Area of inscribed square = a2→ eqn3
Area of circumscribing circle = πR2 where R = radius of circle
⇒ Area of circumscribing circle = πr2→ eqn4
Put the values from equation 3 & 4 in above equation
(from eqn 2)
So, Ratio is π : 2
The ratio is π:2
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.]