Does a circle always have the highest area to perimeter ratio?
For example, if I have 50ft of material to use. I can make a circle shape with a larger area than if I made a square shape with the same perimeter.
If I was to write an equation on maximizing area using a given length, what would it be?

Originally posted by haynewp
Does a circle always have the highest area to perimeter ratio?
For example, if I have 50ft of material to use. I can make a circle shape with a larger area than if I made a square shape with the same perimeter.
If I was to write an equation on maximizing area using a given length, what would it be?


yes, the circle is the curve with the maximum area for a given parameter. it s not too hard to prove, either with a bit of calculus. there isn t really a formula i can give you, but i could outline the proof, if you like....

Wouldn’t the equation be that for the area of a circle, pi * r^2?

no, the area of a circle is (pi*D^2)/4

zanket: nope. read the OP a little more carefully. he s looking for a forumla to tell him which shape has the maximum area for a given perimeter. you ve given him the area for a circle of a given radius

Originally posted by on radioavtives wave
no, the area of a circle is (pi*D^2)/4

That's the same thing as what zanket said.

(pi*D^2)/4 = (pi*D^2)/2^2 = pi*(D/2)^2 = pi * r^2

since 2r=D.

I'm not sure about the formula for maximum area for a certian perimeter though... well, this might work.

Perimeter of length "L".

L=2*pi*r --> r=L/(2*pi)

maximum area is A=pi*r^2 = pi*(L/2*pi)^2 = L^2/(4*pi)

So try that. A=L^2/(4*pi), where L is your perimeter and A is the max area.

Right, that formula won’t do. Since a circle is the shape that has the maximum area for a given perimeter, then wouldn’t the formula be that which returns the area of a circle for the given perimeter, or pi * (c / (2 * pi))^2, where c is the perimeter (or circumference)?

Edit: Orbie beat me to it, and with a simpler formula.

Originally posted by orbie

(pi*D^2)/4 = (pi*D^2)/2^2 = pi*(D/2)^2 = pi * r^2




thanks dude, you just did my homework for me!


now can someone else prove for me that : area of a square = 1/4 peremiter^2 ?

thanks

Given:
L=length of side
Area of square=L^2
Perim of square=L*4

You get:
L=P/4
A=(P/4)^2

i was, of course, kidding

by the way, this question is called the isoperimetric problem in case you want to google, or get a book from the library.

hehe, glad i could help :)

thanks dude, you just did my homework for me!
lol:p

u have to keep all points, on ur perimeter, at equal distance from a point x, without any dent on the perimeter which reduces the available length of the material there by reducing the area too. if the equal distance is at its minimum, u have a circle with x as its centre.;)

I was just goofing around on Autocad when I thought this question. I should have known it already had a name and was solved by someone in the 18th or 19th century.



http://mathworld.wolfram.com/IsoperimetricProblem.html

Originally posted by zanket
Wouldn’t the equation be that for the area of a circle, pi * r^2?

yes

Originally posted by on radioavtives wave
no, the area of a circle is (pi*D^2)/4

yes

wow, the revolution revelation.