Question

i) Four equal circles are described about the four corners of a square so that each circle
touches two of the others. Find the area of the space enclosed between the circumferences
of the circles, each side of the square measuring 24 cm.
​

Answer 1

$\large\sf\underline{Given}$

• $\sf\:Side\:of\:square=24 cm$

$\large\sf\underline{Solution :-}$

We know,

$\tt\red{⋆\:Area\:of\:a\:square=(side)^{2}}$

$\sf\:Area\:of\:a\:square=24^{2}$

$\sf⟹\:Area\:of\:a\:square=24^{2}$

$\tt\purple{⟹\:Area\:of\:a\:square=576 cm^{2}}$

Radius of the 4 circle at corners of square is given by :

$\tt\red{⋆\:Radius=\frac{d}{2}}$

$\sf⟹\:Radius=\frac{24}{2}$

$\tt\purple{⟹\:Radius=12}$

Area of quadrant of circle inside the square is

$\tt\red{=\frac{1}{2} \times π \times r^{2}}$

Since there are 4 such Quadrant :

$\tt\red{⋆\:4 \times \frac{1}{2} \times π \times r^{2}}$

$\sf⟹\:π r^{2}$

$\sf⟹\:π (12) \times 12$

$\sf⟹\:π 144$

$\sf⟹\:3.14 \times 144$

$\tt\purple{⟹\:452.16 cm}$

So the area of remaining portion is :

$\sf\:576-452.16$

$\tt\purple{⟹\:123.84 cm}$

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