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(D) Given that four circular cardboard pieces are placed on a paper such that each piece touches the other two pieces.Let the centers of the four circ

Vedclass · Accepted Answer

$42$

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(D) Given that four circular cardboard pieces are placed on a paper such that each piece touches the other two pieces.Let the centers of the four circles be $A, B, C,$ and $D$. Since the radius of each circle is $7 \, cm$, the distance between the centers of any two touching circles is $7 + 7 = 14 \, cm$.Thus, $AB = BC = CD = DA = 14 \, cm$. This forms a square $ABCD$ with side length $14 \, cm$.The area of the square $ABCD = (side)^2 = (14)^2 = 196 \, cm^2$.Inside the square, there are four sectors, each with a central angle of $90^{\circ}$ and radius $r = 7 \, cm$.The area of one sector $= \frac{90^{\circ}}{360^{\circ}} 	imes \pi r^2 = \frac{1}{4} 	imes \frac{22}{7} 	imes (7)^2 = \frac{1}{4} 	imes 22 	imes 7 = 38.5 \, cm^2$.The total area of the four sectors $= 4 	imes 38.5 = 154 \, cm^2$.The area of the portion enclosed between these pieces = (Area of square $ABCD$) - (Total area of four sectors)$= 196 - 154 = 42 \, cm^2$.Therefore, the required area is $42 \, cm^2$.

Part $I$	Part $II$
$1.$ Formula to find the length of a minor arc	$a.$ $C=2\pi r$
$2.$ Formula to find the area of a minor sector	$b.$ $A=\pi r^{2}$
$3.$ Formula to find the area of a circle	$c.$ $l=\frac{\pi r \theta}{180}$
$4.$ Formula to find the circumference of a circle	$d.$ $A=\frac{\pi r^{2} \theta}{360}$

Four circular cardboard pieces of radii $7 \, cm$ are placed on a paper in such a way that each piece touches the other two pieces. Find the area of the portion enclosed between these pieces (in $cm^2$).

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