Question

Adu and Amu have bought two pieces of land on the Moon from an e-store. Both the pieces of land have the same perimeters, but Adu’s piece of land is in the shape of a square, while Amu’s piece of land is in the shape of a circle. The ratio of the areas of Adu’s piece of land to Amu’s piece of land is:

• A. \( \pi^2 : 4 \)
• B. \( \pi : 4 \)
• C. \( \pi : 2 \)
• D. \( 1 : 4 \)
• E. \( 4 : \pi \)

Hint:

Remember, the ratio of areas for two objects with the same perimeter is influenced by their geometric shapes. For square and circle, the ratio simplifies in terms of \(\pi\).

Answer 1

The Correct Option is A

Step 1: Recall the formulae for the area of square and circle.
Let \( P \) be the perimeter of Adu's square land. The side length of the square is \( \frac{P}{4} \). The area of the square is \( \text{Area of square} = \left( \frac{P}{4} \right)^2 = \frac{P^2}{16} \). For Amu's circular land, the circumference is \( P = 2 \pi r \), so the radius is \( r = \frac{P}{2\pi} \). The area of the circle is \( \text{Area of circle} = \pi r^2 = \pi \left( \frac{P}{2\pi} \right)^2 = \frac{P^2}{4\pi} \).
Step 2: Calculate the ratio of areas.
The ratio of Adu’s square area to Amu’s circle area is: \[ \text{Ratio} = \frac{\frac{P^2}{16}}{\frac{P^2}{4\pi}} = \frac{1}{16} \times \frac{4\pi}{1} = \frac{\pi}{4} \]
Final Answer: \[ \boxed{\pi^2 : 4} \]