section_path: Optimization
section_name: Optimization
paragraph_text: Optimization
  
  Given a wire contour, the surface of least area spanning ("filling") it is a minimal surface. Familiar examples include soap bubbles.
  
  The question of the filling area of the Riemannian circle remains open.
  
  The circle has the largest area of any two-dimensional object having the same perimeter.
  
  A cyclic polygon (one inscribed in a circle) has the largest area of any polygon with a given number of sides of the same lengths.
  
  A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral.
  
  The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral.
  
  The ratio of the area of the incircle to the area of an equilateral triangle, π 3 3 {\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}, is larger than that of any non-equilateral triangle.
  
  The ratio of the area to the square of the perimeter of an equilateral triangle, 1 12 3, {\displaystyle {\frac {1}{12{\sqrt {3}}}},} is larger than that for any other triangle.