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: Understanding the Concept:
The shaded area can be found by calculating the area of the square ABCD and subtracting the areas of the four circular sectors that are inside the square. Step 2: Key Formula or Approach:
1. Determine the radius of the circles.
2. Calculate the area of the square.
3. Calculate the area of the four sectors inside the square.
4. Area of Shaded Region = Area of Square - Area of 4 Sectors. Step 3: Detailed Explanation:
The side of the square ABCD is given as 14 cm. Since the circles with centers A and B (or A and D) touch externally, the side of the square is the sum of the radii of two circles.
Side of square = radius + radius = 2 \( \times \) radius.
\( 14 = 2r \)
\( r = \frac{14}{2} = 7 \) cm.
The radius of each circle is 7 cm.
Area of the square ABCD = \( (\text{side})^2 = (14)^2 = 196 \) cm\(^2\).
The four sectors inside the square are at the corners. Since ABCD is a square, the angle of each corner is 90\(^{\circ}\). So, each sector has a central angle of 90\(^{\circ}\).
The sum of the angles of the four sectors is \( 4 \times 90^{\circ} = 360^{\circ} \), which is the angle of a complete circle.
Therefore, the combined area of the four sectors is equal to the area of one full 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\(^2\).
Now, calculate the area of the shaded region:
Area of Shaded Region = Area of Square - Area of 4 Sectors
Area of Shaded Region = \( 196 - 154 = 42 \) cm\(^2\). Step 4: Final Answer:
The area of the shaded region is 42 cm\(^2\).
