Analogy — Match the boundary measure to the shape: Circle : Circumference :: Square : ?

Introduction / Context:
In analogies, we map the relation in the first pair onto the second pair. “Circle : Circumference” associates a geometric figure with the length of its boundary. We must choose the square’s corresponding boundary measure.

Given Data / Assumptions:

• For a circle, circumference is the total boundary length around the circle.
• For a square, the comparable boundary quantity is its perimeter.
• Other geometric quantities (area, diagonal, volume) are different attributes, not boundary length.

Concept / Approach:
Identify the relation “figure → boundary-length measure.” Preserve type (1-D boundary), not 2-D (area) or 3-D (volume). For polygons the boundary length is called perimeter; for circles it is circumference.

Step-by-Step Solution:
Map: circle → circumference (boundary length).Seek: square → ? (boundary length) = perimeter.Therefore, “Square : Perimeter.”

Verification / Alternative check:
Units: circumference and perimeter both measure length (1-D). Area measures square units; diagonal is a single segment length inside the shape; volume applies to solids, not plane figures. Only “perimeter” matches the boundary-length concept for a polygon.

Why Other Options Are Wrong:
(a) Volume – not defined for plane square; needs 3-D solid.
(b) Area – 2-D measure; does not parallel boundary length.
(c) Diagonal – a specific internal segment; not the complete boundary length.

Common Pitfalls:
Confusing any well-known formula of a square (area or diagonal) with the boundary measure. Analogy problems require matching the type of attribute, not just any attribute of the second figure.

Final Answer:
Perimeter