The ratio of the length to the breadth of a rectangle is $3:2$ and its area is $216 ext{ sq. cm}$. If the circumference of a circle is $28 ext{ cm}$ more than the perimeter of the rectangle, find the area of the circle (in sq. cm).

Question

The ratio of the length to the breadth of a rectangle is $3:2$ and its area is $216 	ext{ sq. cm}$. If the circumference of a circle is $28 	ext{ cm}$ more than the perimeter of the rectangle, find the area of the circle (in sq. cm).

Accepted Answer

**Solution:**

1. Let the length and breadth of the rectangle be $3x$ and $2x$ respectively.

2. Given the area of the rectangle:
   $$3x 	imes 2x = 6x^2 = 216$$
   $$\Rightarrow x^2 = \frac{216}{6} = 36$$
   $$\Rightarrow x = 6$$

3. Therefore, length = $3 	imes 6 = 18 	ext{ cm}$ and breadth = $2 	imes 6 = 12 	ext{ cm}$.

4. Calculate the perimeter of the rectangle:
   $$P = 2(	ext{length} + 	ext{breadth}) = 2(18 + 12) = 2 	imes 30 = 60 	ext{ cm}$$

5. The circumference of the circle is $28 	ext{ cm}$ more than the perimeter of the rectangle:
   $$C = 60 + 28 = 88 	ext{ cm}$$

6. Circumference of a circle is given by:
   $$C = 2 \pi r$$
   $$\Rightarrow 2 \pi r = 88$$
   $$\Rightarrow r = \frac{88}{2 \pi} = \frac{44}{\pi}$$

7. Area of the circle:
   $$A = \pi r^2 = \pi \left(\frac{44}{\pi}ight)^2 = \pi 	imes \frac{1936}{\pi^2} = \frac{1936}{\pi}$$

8. Using $\pi \approx 3.14$:
   $$A \approx \frac{1936}{3.14} \approx 616.56$$

**Answer:** $616 	ext{ sq. cm}$

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