Maximizing Area: Perimeter of 2D Figures and Volume of 3D Figures Explained

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SUMMARY

The discussion centers on the mathematical principle that among two-dimensional figures with a fixed perimeter, the circle maximizes area, while among three-dimensional figures with a fixed volume, the sphere maximizes surface area. This conclusion is derived from solving differential equations that describe the relationship between area and perimeter in 2D and volume and surface area in 3D. The complete mathematical framework involves the "Calculus of Variations," which provides the necessary tools to derive these results definitively.

PREREQUISITES
  • Understanding of differential equations
  • Familiarity with the concept of perimeter and area in geometry
  • Knowledge of surface area and volume in three-dimensional geometry
  • Basic principles of Calculus of Variations
NEXT STEPS
  • Study the principles of Calculus of Variations
  • Explore differential equations related to geometric optimization
  • Investigate the relationship between perimeter and area in various 2D shapes
  • Examine the properties of spheres and their surface area calculations
USEFUL FOR

Mathematicians, physics students, geometry enthusiasts, and anyone interested in optimization problems in two and three dimensions.

Amith2006
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Homework Statement


For a two DIMENSIONAL figure of given perimeter why does the circle have the largest area? Similarly, of all three dimensional figures of given volume the sphere has the largest area. why?

Homework Equations





The Attempt at a Solution

 
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You can set up a differential equation describing the relationship between area and perimeter for varying shapes... solving for the maximum results in the equation for a circle. The same thing in 3D results in a sphere.
 
thanx buddy.
 
The complete answer involves "Calculus of Variations".
 

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