This problem is talked about in the topic of calculus of variation. For someone who is not familiar with this area of math, how can we prove it?

I thought this was equivalent to proving that among the shapes with fixed area, a circle has the least perimeter.

Then I thought that we could consider a circle, then consider an infinitesimal arc of it subtending an infinitesimal angle. Then we could say that if you try to alter the arc at that point, the perimeter will get larger. But the area will not change much (because the infinitesimal rank of area is 2 but that of a line segment is 1). So it means that circle must have the least perimeter.

But then I came up with the conclusion that this can be applied to any kind of curve (more precisely, not just circle). So this solution will not work.

Is it possible? Any ideas?