Four circles of equal radius are drawn with centers, A, B, C and D such that ABCD is a square of side 14 cm and the circles touch externally as in the figure. The area of the shaded region bounded by the 4 circles is: (Take \( \pi = \frac{22}{7} \))
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Whenever you see a problem with four identical sectors at the corners of a square, remember that their combined area is equal to the area of one full circle with the same radius. The calculation simplifies to Area of Square - Area of Circle.
Step 1: Concept Identification: The shaded area is derived by calculating the area of square ABCD and subtracting the areas of the four circular sectors within the square. Step 2: Method Outline: 1. Ascertain the radius of the circles. 2. Compute the area of the square. 3. Calculate the area of the four sectors situated inside the square. 4. Shaded Region Area = Square Area - Area of 4 Sectors. Step 3: Detailed Derivation: The side length of square ABCD is provided as 14 cm. Given that the circles centered at A and B (or A and D) are externally tangent, the square's side length equals the sum of two radii. Square Side = radius + radius = 2 \( \times \) radius. \( 14 = 2r \) \( r = \frac{14}{2} = 7 \) cm. Each circle has a radius of 7 cm. Area of Square ABCD = \( (\text{side})^2 = (14)^2 = 196 \) cm\(^2\). The four sectors are positioned at the corners of the square. As ABCD is a square, each corner angle is 90\(^{\circ}\). Thus, each sector has a central angle of 90\(^{\circ}\). The aggregate angle of the four sectors is \( 4 \times 90^{\circ} = 360^{\circ} \), equivalent to a full circle. Consequently, the combined area of the four sectors equals the area of a single circle with radius r = 7 cm. Area of 4 Sectors = Area of One Circle = \( \pi r^2 \). Area of 4 Sectors = \( \frac{22}{7} \times (7)^2 = \frac{22}{7} \times 49 = 22 \times 7 = 154 \) cm\(^2\). Calculation of the shaded region's area: Area of Shaded Region = Area of Square - Area of 4 Sectors Area of Shaded Region = \( 196 - 154 = 42 \) cm\(^2\). Step 4: Conclusive Result: The area of the shaded region is 42 cm\(^2\).
