My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Reply
 
LinkBack Thread Tools Display Modes
January 31st, 2015, 01:29 AM   #1
Newbie
 
Joined: Aug 2012

Posts: 22
Thanks: 0

Perimeter and area of the circle

How to prove that a circle has the maximum possible area for a given perimeter, and the minimum possible perimeter for a given area.
matemagia is offline  
 
January 31st, 2015, 05:17 AM   #2
Senior Member
 
Olinguito's Avatar
 
Joined: Apr 2014
From: Greater London, England, UK

Posts: 320
Thanks: 155

Math Focus: Abstract algebra
This is the so called isoperimetric problem.

Quote:
Although the circle appears to be an obvious solution to the problem, proving this fact is rather difficult. The first progress toward the solution was made by Swiss geometer Jakob Steiner in 1838, using a geometric method later named Steiner symmetrisation. Steiner showed that if a solution existed, then it must be the circle. Steiner's proof was completed later by several other mathematicians.

Steiner begins with some geometric constructions which are easily understood; for example, it can be shown that any closed curve enclosing a region that is not fully convex can be modified to enclose more area, by "flipping" the concave areas so that they become convex. It can further be shown that any closed curve which is not fully symmetrical can be "tilted" so that it encloses more area. The one shape that is perfectly convex and symmetrical is the circle, although this, in itself, does not represent a rigorous proof of the isoperimetric theorem.
Olinguito is offline  
January 31st, 2015, 06:06 AM   #3
Math Team
 
Joined: Jan 2015
From: Alabama

Posts: 3,261
Thanks: 894

A general proof would require "Calculus of Variations"- Calculus of Variations -- from Wolfram MathWorld. Are you familiar with that?
Country Boy is offline  
January 31st, 2015, 06:39 AM   #4
Global Moderator
 
Joined: Dec 2006

Posts: 19,888
Thanks: 1836

See .
skipjack is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
area, circle, perimeter



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
find area and perimeter Albert.Teng Algebra 5 July 27th, 2012 08:25 AM
Perimeter & Area Optimization Chee Calculus 1 April 1st, 2012 05:47 PM
area and perimeter tsl182forever8 Algebra 1 February 28th, 2012 11:58 PM
Perimeter of a triangle circumscribed about a circle? BigB7 Algebra 1 January 18th, 2011 06:04 PM
Given perimeter, two fields, max area and a cow frankiee Algebra 7 February 6th, 2010 08:39 PM





Copyright © 2018 My Math Forum. All rights reserved.