section_path: Analytic results/Area enclosed
section_name: Area enclosed
paragraph_text: Area enclosed
  
  As proved by 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 and whose height equals the circle's radius, which comes to π multiplied by the radius squared: A r e a = π r 2. {\displaystyle \mathrm {Area} =\pi r^{2}.}
  
  Equivalently, denoting diameter by d, A r e a = π d 2 4 ≈ 0.7854 d 2, {\displaystyle \mathrm {Area} ={\frac {\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.