Is the perimeter always smaller than the area?

Answer 1

The perimeter is in linear units (e.g. meters), and the area in square units (e.g. square meters), so you can't compare them directly.

If you insist on comparing a square unit with a linear unit - even though this has no physical significance! - it all depends on the units chosen, and the size.

A square of 1x1 has a surface area of 1 square unit, but a perimeter of 4 units. This is a counterexample to your proposition. At a size of 4x4, you reach the "break-even point"; above that, the perimeter would have a lower numerical value than the area. But please note that if you use physical measurements, the square of 1 meter x 1 meter (for example) has a perimeter of 4 meters and an area of 1 square meter (perimeter has a higher numerical value), but when you change units to centimeters, the same square has a perimeter of 400 cm, and an area of 10,000 cubic centimeters (here, the perimeter has a LOWER numerical value).