Question

Is it possible for two squares to have the same area without having the same perimeter?

Answer 1

Explanation:

if that were the case, we would have for the areas

a² = b²

but for the perimeters

4a <> 4b

a <> b

and therefore

a² <> b²

yet,

a² = b²

AND

a² <> b²

at the same time is impossible.

also from

a² = b²

we get

|±sqrt(a²)| = |±sqrt(b²)|

and |±a| = |±b|

the only wiggle room would be that they have different signs.

but that would mean a or b must be negative.

but a negative side length for a square does not make any sense.

so, no matter how we look at it, if the area is the same, it means that the side lengths must be the same. and therefore also the perimeter.

no, it is not possible for 2 squares to have the same area but not the same perimeter.