As proved by my good friend Archimedes,
in his Measurement of a Circle,
the area enclosed by a circle is equal
to that of a triangle whose base has the length
of the circle's circumference &
whose height equals the circle's radius,
which comes to π multiplied by the radius squared:

Area = pi r^2.
Equivalently, denoting diameter by d

Area =pi d^2/4 approx 0.7854d^2,
that is,                               approximately
          79% of the circumscribing
square whose side is of length d

The circle is the plane curve enclosing
the maximum area for a given arc length.
This relates the circle to a problem
in the calculus of variations,
namely the isoperimetric inequality [of course]