The area of a square field is 69696 square centimetres. What is the length of its diagonal, in centimetres?

Introduction:
This geometry question involves the relationship between the area, side length, and diagonal of a square. The area of the square is given, and we are asked to find the diagonal. This requires first finding the side length from the area and then using the well-known relationship between the side and diagonal of a square. The problem tests comfort with square roots and the Pythagorean-type relation for squares.

Given Data / Assumptions:

• Area of the square field = 69696 sq cm. • Let the side length of the square be s cm. • Area of a square = s^2. • Diagonal d of a square with side s = s * √2.

Concept / Approach:
Since the area of a square is s^2, we can find s by taking the square root of the given area. Once we know s, we use the fact that the diagonal of a square forms the hypotenuse of a right-angled triangle with both legs equal to s. By the Pythagorean theorem, diagonal d satisfies d^2 = s^2 + s^2 = 2s^2, so d = s * √2. After computing s, we multiply by an approximate value of √2 (or keep in mind approximate products) to find the diagonal.

Step-by-Step Solution:
Step 1: Use the area formula to find the side length. Area = s^2 = 69696 sq cm. Step 2: Take the square root of 69696. s = √69696. Step 3: Compute √69696. 69696 = 264^2, so s = 264 cm. Step 4: Use the diagonal formula for a square. d = s * √2 = 264 * √2. Step 5: Approximate √2 ≈ 1.414. d ≈ 264 * 1.414. Step 6: Multiply 264 by 1.414. 264 * 1.414 ≈ 373.35 cm (approximately). Step 7: Choose the nearest option. The nearest option to 373.35 cm is 373 cm.

Verification / Alternative check:
We can verify by squaring 373 to see if it is close to the theoretical diagonal squared. If d ≈ 373.35 cm, then d^2 ≈ 139,384. For 373^2, we get 139,129, which is close enough to the computed 2 * 69696 = 139,392, given rounding of √2. This confirms that 373 cm is a very good approximation for the diagonal and matches the nearest option provided.

Why Other Options Are Wrong:
Option 473 cm: This corresponds to a much larger diagonal that would produce a side longer than 264 cm, contradicting the given area. Option 573 cm and 673 cm: These values are even larger and would lead to side lengths that make the area far greater than 69696 sq cm. Option 273 cm: This is too small for the diagonal of a square with side 264 cm, because the diagonal must be longer than any side.

Common Pitfalls:
Some students may mistakenly take the square root of the area to be the diagonal instead of the side, or they may forget to multiply the side by √2. Others may approximate √2 too coarsely or round intermediate results incorrectly. It is also possible to make arithmetic mistakes while finding the exact square root of 69696. Writing each step clearly and using known squares (like 264^2) helps avoid confusion.

Final Answer:
The length of the diagonal of the square field is approximately 373 cm.