Question

An interschool rangoli competition was organised by one of the reputed schools of Odisha. The theme of the rangoli competition was Diwali celebrations, wherein students were supposed to make mathematical designs. Students from various schools participated and made beautiful rangoli designs. One such design is shown in the given figure. Rangoli is in the shape of a square ABCD of side 40 cm. At each corner of the square, a quadrant of a circle of radius 10 cm is drawn in which diyas are kept. A circle of diameter 20 cm is also drawn inside the square.

Based on the above information, answer the following questions.

1. Find the area of the square ABCD.
2. What is the area of the circle drawn inside the square?
3. If the circle and the four quadrants are cut off from the square ABCD and removed then find the area of the remaining portion of the square ABCD.
4. Find the combined area of the four quadrants and the circle removed.

Answer 1

i. Area of square ABCD

Side of the square a = 40 cm.

\[\text{Area} = a^2\]

\[= 40 \times 40\]

\[= 1600\ \text{cm}^2\]

ii. Area of the circle inside the square

Diameter = 20 cm, so radius \(R = \dfrac{20}{2} = 10\) cm.

\[\text{Area} = \pi R^2\]

\[= \frac{22}{7} \times 10 \times 10\]

\[= \frac{2200}{7} \approx 314.29\ \text{cm}^2\]

iii. Area of the remaining portion

Radius of each quadrant r = 10 cm.

\[\text{Combined area of 4 quadrants} = 4 \times \frac{1}{4}\pi r^2\]

\[= \pi r^2\]

\[= \frac{22}{7} \times 10 \times 10\]

\[= \frac{2200}{7}\ \text{cm}^2\] 

\[\text{Remaining area} = \text{Area of square} - \left( \text{Area of circle} + \text{Area of 4 quadrants} \right)\] 

\[= 1600 - \left( \frac{2200}{7} + \frac{2200}{7} \right)\]

\[= 1600 - \frac{4400}{7}\]

\[= \frac{6800}{7} \approx 971.43\ \text{cm}^2\]

iv. Combined area of the four quadrants and the circle removed

\[= \frac{2200}{7} + \frac{2200}{7}\]

\[= \frac{4400}{7} \approx 628.57\ \text{cm}^2\]