A = Area of the largest circle drawn inside a square of side 1 cm
B = Sum of areas of 4 identical (largest possible) circles drawn inside a square of side 1 cm.
C = Sum of areas of 9 identical circle (largest possible) drawn inside a square of side 1 cm.
D = Sum of area of 16 identical circles (largest possible) drawn inside a square of side 1 cm. Which of the following is TRUE abour A, B, C and D ?

Options

A. `A gt B gt C gt D`
B. `A lt B lt C lt D`
C. `A gt B = C gt D`
D. No option is correct

Answer

To solve the problem, we need to calculate the areas of the largest circles that can be drawn inside a square of side 1 cm for each case (A, B, C, and D).

### Step-by-Step Solution:

1. **Calculate Area A:**
- The largest circle that can be drawn inside a square has a diameter equal to the side of the square.
- Given that the side of the square is 1 cm, the diameter of the circle is also 1 cm.
- The radius \( r_A \) of the circle is half of the diameter:
\[ r_A = \frac{1}{2} \text{ cm} \]
- The area \( A \) of the circle is given by the formula:
\[ A = \pi r_A^2 = \pi \left(\frac{1}{2}\right)^2 = \pi \cdot \frac{1}{4} = \frac{\pi}{4} \text{ cm}^2 \]

2. **Calculate Area B:**
- For 4 identical circles, the largest possible circles that can fit inside the square will have a diameter equal to half the side of the square (since they are arranged in a 2x2 grid).
- Thus, the diameter of each circle is:
\[ \text{Diameter} = \frac{1}{2} \text{ cm} \]
- The radius \( r_B \) is:
\[ r_B = \frac{1}{4} \text{ cm} \]
- The area of one circle is:
\[ \text{Area of one circle} = \pi r_B^2 = \pi \left(\frac{1}{4}\right)^2 = \pi \cdot \frac{1}{16} = \frac{\pi}{16} \text{ cm}^2 \]
- Therefore, the total area \( B \) for 4 circles is:
\[ B = 4 \times \frac{\pi}{16} = \frac{4\pi}{16} = \frac{\pi}{4} \text{ cm}^2 \]

3. **Calculate Area C:**
- For 9 identical circles, they can be arranged in a 3x3 grid.
- The diameter of each circle is:
\[ \text{Diameter} = \frac{1}{3} \text{ cm} \]
- The radius \( r_C \) is:
\[ r_C = \frac{1}{6} \text{ cm} \]
- The area of one circle is:
\[ \text{Area of one circle} = \pi r_C^2 = \pi \left(\frac{1}{6}\right)^2 = \pi \cdot \frac{1}{36} = \frac{\pi}{36} \text{ cm}^2 \]
- Therefore, the total area \( C \) for 9 circles is:
\[ C = 9 \times \frac{\pi}{36} = \frac{9\pi}{36} = \frac{\pi}{4} \text{ cm}^2 \]

4. **Calculate Area D:**
- For 16 identical circles, they can be arranged in a 4x4 grid.
- The diameter of each circle is:
\[ \text{Diameter} = \frac{1}{4} \text{ cm} \]
- The radius \( r_D \) is:
\[ r_D = \frac{1}{8} \text{ cm} \]
- The area of one circle is:
\[ \text{Area of one circle} = \pi r_D^2 = \pi \left(\frac{1}{8}\right)^2 = \pi \cdot \frac{1}{64} = \frac{\pi}{64} \text{ cm}^2 \]
- Therefore, the total area \( D \) for 16 circles is:
\[ D = 16 \times \frac{\pi}{64} = \frac{16\pi}{64} = \frac{\pi}{4} \text{ cm}^2 \]

### Summary of Areas:
- \( A = \frac{\pi}{4} \)
- \( B = \frac{\pi}{4} \)
- \( C = \frac{\pi}{4} \)
- \( D = \frac{\pi}{4} \)

### Conclusion:
Since all areas A, B, C, and D are equal, we can conclude that:
- **A = B = C = D**