If the diagonal of a square is doubled, how does the area of the square change?

Introduction / Context:
The area of a square is related to its diagonal by A = (d^2)/2. Thus, the diagonal serves as a direct linear measure whose square is proportional to area. Doubling the diagonal will therefore square the scale factor for area, leading to a fourfold change.

Given Data / Assumptions:

• Original diagonal = d
• New diagonal = 2d
• Area formula in terms of diagonal: A = d^2 / 2

Concept / Approach:
Compare new area A′ = ( (2d)^2 ) / 2 with old area A = d^2 / 2. Take the ratio A′/A to find the multiplicative change in area after doubling the diagonal.

Step-by-Step Solution:
A = d^2 / 2A′ = (2d)^2 / 2 = 4d^2 / 2 = 2d^2A′ / A = (2d^2) / (d^2 / 2) = 4

Verification / Alternative check:
Let a square have side s; then diagonal d = s√2 and area A = s^2. Doubling d implies the new side is doubled (since s = d/√2), giving area (2s)^2 = 4s^2 = 4A — same conclusion.

Why Other Options Are Wrong:
Twofold and threefold changes would correspond to diagonal scale factors of √2 and √3, not 2. “None of the above” is incorrect since “four fold” is exactly right.

Common Pitfalls:
Confusing diagonal doubling with side doubling via a mistaken linear relation. Because area depends on the square of linear scale, doubling any linear dimension multiplies area by 4.

Final Answer:
Becomes four fold