If the diagonal of a square is doubled, then the area of the square will be:

For a square, the area can be written directly in terms of its diagonal d: Area = d2/2 (since side = d/√2, so side2 = d2/2).

1. Let the original diagonal be d, so the original Area = d2/2.

2. When the diagonal is doubled, the new diagonal becomes 2d.

3. New Area = (2d)2/2 = 4d2/2 = 4 × (d2/2).

4. Since d2/2 is exactly the original Area, the New Area = 4 × Original Area.

Cross-check with a concrete case: take side = 1 unit, so diagonal = √2 and Area = 1. Doubling the diagonal to 2√2 gives a new side of 2, so the new Area = 22 = 4 — exactly four times the original Area of 1, confirming the scaling independently.

So doubling the diagonal makes the area four times the original — not three times, the same, or eight times.