A square pond has 2 m sides and is 1 m deep. If it is to be enlarged, the depth remaining the same, into a circular pond with the diagonal of the square as diameter as shown in the figure, then what would be the volume of earth to be removed?

### Concept & Volume Difference The volume of earth to be removed is the difference between the final volume of the new shape and the initial volume of the existing shape. $$ \text{Volume Removed} = V_{\text{final}} - V_{\text{initial}} $$ ### Step-by-Step Solution * Given for original square pond: side $a = 2 \text{ m}$, depth $h = 1 \text{ m}$. * Initial Volume $V_{\text{initial}} = \text{Area of base} \times \text{depth} = (2 \times 2) \times 1 = 4 \text{ m}^3$. * The diagonal of the square is $a\sqrt{2} = 2\sqrt{2} \text{ m}$. * The new circular pond has this diagonal as its diameter. So, radius $r = \sqrt{2} \text{ m}$. Depth remains $h = 1 \text{ m}$. * Final Volume $V_{\text{final}} = \pi \times r^2 \times h = \pi \times (\sqrt{2})^2 \times 1 = 2\pi \text{ m}^3$. * Volume of earth removed = $V_{\text{final}} - V_{\text{initial}} = (2\pi - 4) \text{ m}^3$. ### Exam Strategy & Shortcut Recognize that a circle circumscribing a square has double the area of the square inscribed in it, but here the diagonal is the diameter. Area of square = $a^2 = 4$. Area of circle with diameter $a\sqrt{2}$ is $\frac{\pi (2\sqrt{2})^2}{4} = 2\pi$. Since depth is 1, volume difference is simply $2\pi - 4$. ### Common Pitfall Using the side of the square as the diameter of the circle instead of the diagonal, which would lead to an incorrect final volume. ### Final Answer Therefore, the correct answer is **$(2\pi - 4) \text{ m}^3$**.