# MathsWatch

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#1
ok its the farmers fencing problem, where the farmer has 1000 metres of fencing, and a circle is meant to be the biggest area, but in mine the polygon with 1000 sides is bigger (and everything with more sides than that too).

so to work out area for the polygons it is:
1/2 (500/n)((500/n)/tan(360/2))2n

where n= number of sides the polygon has

and to work out the area of circle
find the radius from a circumference of 1000m
so;
2pi

then to find the area it is; 2pi r^2

but it comes out that the circle is smaller than the polygons, and i know this is wrong, pls someone help!
0
16 years ago
#2
(Original post by Me2)
ok its the farmers fencing problem, where the farmer has 1000 metres of fencing, and a circle is meant to be the biggest area, but in mine the polygon with 1000 sides is bigger (and everything with more sides than that too).

so to work out area for the polygons it is:
1/2 (500/n)((500/n)/tan(360/2))2n

where n= number of sides the polygon has

and to work out the area of circle
find the radius from a circumference of 1000m
so;
2pi

then to find the area it is; 2pi r^2

but it comes out that the circle is smaller than the polygons, and i know this is wrong, pls someone help!
For the area of an n sided polygon, I got:

A = ( 500^2 )/( n*tan(pi/n) )

If you are unfamiliar with radians, I will just say that tan(pi/n) is equivilent to tan(180/n).

Using this formula with n = 1000, you get:

A = 79577.20975

But, if you consider the circle with circumference 1000. You have:

2pi*r = 1000
r^2 = ( 500/pi )^2

So, A = pi*r^2 = (500^2)/pi = 79577.47155

Which is 0.2617996 bigger than the area of a 1000-sided polygon, which, if you think about it kind of makes sense because a 1000-sided polygon is very close to a circle.
0
#3
(Original post by mikesgt2)
For the area of an n sided polygon, I got:

A = ( 500^2 )/( n*tan(pi/n) )
but why is it 500^2 ?
0
16 years ago
#4
(Original post by Me2)
but why is it 500^2 ?
Look at the attatchment, it shows a segment of an n-sided polygon. It is a made up of two triagles each with base b and height a and with an angle pi/n.

The area of one of the trangles is ab/2, so the area of the segment is ab. Therefore, we have as the area of the whole n-sided polygon, which is made up of n segments:

A = nab

Now, the perimeter of this shape is 2nb. So, given that the perimeter must be 1000 we have:

2nb = 1000
nb = 500

Sub this into the equation for the area:

A = 500a

Also, by trigonometry we have:

tan(pi/n) = b/a
a = b/tan(pi/n)

But from above b = 500/n, so:

a = 500/( n*tan(pi/n) )

So, substituting this expression for a gives:

A = 500a = 500( 500/( n*tan(pi/n) ) ) = ( 500^2 )/( n*tan(pi/n) )

So that is where the 500^2 comes from.
0
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