- 29 Dec '05 21:19A gardner wants to fence in an area of land with a fence. He wants to fence in as much land as possible with as little fence as possible. He is a neat freak so the plot of land he fences in has to be symmetrical.

He goes to the local mega-store and finds that he can purchase fence in 1-meter lenghts. He calculates that if he purchases 3 lengths and joins them together in an equilaterial triangle the area encompassed will be 0.433 m^2, with an effective use of 0.144 m^2/m of fence. He then calculates further that if he purchases 4 lengths he can enclose 1 m^2 for an effective use of 0.25 m^2/m.

Not being a great mathematician (unlike many of the folks on this website) he is unable to caculate the point at which the ratio of area to perimeter begins to go down, so that he can then determine the most effective fencing usage (i.e., where the area enclosed per meter of fence used is the greatest). Can you help him out? - 29 Dec '05 22:06

Area of a regular Polygon is given by:*Originally posted by The Plumber***A gardner wants to fence in an area of land with a fence. He wants to fence in as much land as possible with as little fence as possible. He is a neat freak so the plot of land he fences in has to be symmetrical.**

He goes to the local mega-store and finds that he can purchase fence in 1-meter lenghts. He calculates that if he purchases 3 lengths a ...[text shortened]... (i.e., where the area enclosed per meter of fence used is the greatest). Can you help him out?

A = n*s^2 /4*tan(180/n)

where n is the number of sides and s is the side length.

As perimeter is proportional to number of sides we can then divide by n again and find a maximum.

R = s^2 / 4*tan(180/n)

Differeniting this gives

dR/dn = 1/4*s^2*(1+tan(pi/n)^2)*pi/(tan(pi/n)^2*n^2)

dR/dn will be zero when (1+tan(pi/n)^2) is zero). This however makes no sense. - 29 Dec '05 22:32

It doesn't make sense because the shape with the highest area/perimeter ratio is a circle, and as more sides are added the more the fence resembles a circle.*Originally posted by XanthosNZ***Area of a regular Polygon is given by:**

A = n*s^2 /4*tan(180/n)

where n is the number of sides and s is the side length.

As perimeter is proportional to number of sides we can then divide by n again and find a maximum.

R = s^2 / 4*tan(180/n)

Differeniting this gives

dR/dn = 1/4*s^2*(1+tan(pi/n)^2)*pi/(tan(pi/n)^2*n^2)

dR/dn will be zero when (1+tan(pi/n)^2) is zero). This however makes no sense. - 29 Dec '05 22:42

That would be fine except the question isn't asking for greatest proportion of a circle it's asking for the maximum of A/p (or A/n makes no difference).*Originally posted by richjohnson***It doesn't make sense because the shape with the highest area/perimeter ratio is a circle, and as more sides are added the more the fence resembles a circle.** - 29 Dec '05 23:24

What I was trying to say (albeit in a roundabout way) is that I do not believe there is a maximum. Increasing the number of sides increases the A/p ratio, although I suppose there is a point where the gain is so marginal that the extra side is not worth it.*Originally posted by XanthosNZ***That would be fine except the question isn't asking for greatest proportion of a circle it's asking for the maximum of A/p (or A/n makes no difference).** - 06 Nov '06 11:38

The answer is simple - it's a circle.*Originally posted by The Plumber***A gardner wants to fence in an area of land with a fence. He wants to fence in as much land as possible with as little fence as possible. He is a neat freak so the plot of land he fences in has to be symmetrical.**

He goes to the local mega-store and finds that he can purchase fence in 1-meter lenghts. He calculates that if he purchases 3 lengths a ...[text shortened]... (i.e., where the area enclosed per meter of fence used is the greatest). Can you help him out? - 06 Nov '06 14:46The story ends like this (my version):

After a while experimenting he found out to ask a very intelligent, bright man (who happened to be a chess player) of the name Fabian. Fabian put up the fence and proudly demonstrated the solution from the middle of the polygon, saying: "This is the solution!"

The farmer said: "This is not a big area you've fenced in? No much place for more than on cow!"

Fabian said with pride: "Oh no, I think the very way around. What you see is not inside of the fence. The area I've fenced in is the rest of the entire world! This small polygon in the middle is what's left of the world!

It's only a definition of what's inside and what's outside. - 06 Nov '06 15:51

Just a thought - if you don't want a circle -- the next best thing would be a polygon with as many sides as is pratically possible.*Originally posted by sugiezd***Now, don't get huffy.**

Plastic fencing can be bent to form a perfect circle.

Define you question more carefully next time.

After all, would a polygon with an infinite number of sides not be a circle?