Question

ABCD is a square of side 14 cm .
With centres A, B , C and D , 4 circles r drawn such that each circle touch externally 2 of the remaining 3 circles.
Find the area of the shaded region.


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Answer 1

Question

In figure, ABCD is a square of side 14cm. With centres A, B, C and D, four circles are drawn such that each circle touch externally two of the remaining three circles. Find the area of the shaded region.

To find

Area of shaded region

Given

Side of square ABCD = 14 cm

Radius of circles with centers A, B, C and D = 14/2 = 7 cm

Solution

Area of shaded region = Area of square - Area of four sectors subtending right angle

Area of each of the 4 sectors is equal to each other and is a sector of 90° in a circle of 7 cm radius. So, Area of four sectors will be equal to Area of one complete circle

So

$area \: of \: 4 \: sectors = \frac{22}{7 } \times 7 \times 7$

$area \: of \: 4 \: sectors \: = 154 {cm}^{2}$

Area of 4 sectors = Πr²

Area of square ABCD = (Side)²

Area of square ABCD = (14)²

Area of square ABCD = 196 cm²

Area of shaded portion = Area of square ABCD - 4 × Area of each sector

= 196 – 154

= 42 cm²

Therefore, the area of shaded portion is 42 cm²

$bts ❤ exo$

Answer 2

Answer:

$\bf \fbox \red { AREA of the shaded region is 42 cm² }$

Step-by-step explanation:

Side of square ABCD = 14 cm

$Radius \: o f \: each \: quadrant \: of \: a \: circle = \frac{14}{2} \\ = 7 \: cm$

$Area \: of \: all \: 4 \: quadrant \: of \: circles = 4 \times \frac{1}{4} \pi \: {r}^{2} \\ \\ = \frac{22}{7} \times 7 \times 7 \\ \\ = 154 \: {cm}^{2}$

$AREA \: of \: square \: ABCD = a² \\ \\ = 14 \times 14 \\ \\ = 196 \: {cm}^{2}$

$Area \: of \: the \: shaded \: region = 196 - 154 \\ \\ = 42 \: {cm}^{2}$

$\bf \fbox \purple{ Hence the area of the shaded region is 42 cm² }$