Phil Bavister
The Fencing Problem
A farmer has exactly 100 metres of fencing and wants to fence off a plot of land.
She is not concerned about the shape of the plot, but it must have a perimeter of 1000 m.
She wishes to fence off the plot of land, which contains the maximum area.
Investigate the shape, or shapes, that could be used to fence in the maximum area using exactly 1000 m of fencing each time.
Firstly I thought about some of the shapes the possible and decided to calculate their areas in a logical order. So I decided to start with triangles before moving onto quadrilaterals, pentagons, hexagons and so on.
Triangles
To calculate the area of a triangle you must times half the base by the vertical height.
Formulae
A = b x h
Equilateral Triangle
There is more than one way to calculate the area of an equilateral triangle so I will demonstrate each of the three different ways.
. A = x x x sin 60
= 48112.52 (to 2d.p.)
2. A = x - 2 x h
h = sin 60 - - 3
= 48112.52 (to 2d.p.)
3. A = ab sin c
= 48112.52 (to 2d.p.)
Isosceles Triangle
A = b x h
To find the vertical height you must split the isosceles triangle into two separate right-angled triangles and use Pythagoras' Theorem. Pythagoras can only be used in right-angled triangles and uses two sides to find the third side.
h = a2 - (b)sq. = Ans.
h =
Formulae
A = b x
Examples
200 x (90000 - 40000 = 50000.
) = 44721.36 (to 2d.p.)
00 x (160000 - 1000 = 150000
) = 38729.83 (to 2d.p.)
Length of Sides (m)
a
b
c
Height (m)
Area (m )
251
498
251
31.62278
7874.071
275
450
275
58.1139
35575.62
300
400
300
223.6068
44721.36
325
350
325
273.8613
47925.72
330
340
330
282.8427
48083.26
333
334
333
288.0972
48112.23
335
330
335
291.5476
48105.35
340
320
340
300
48000
345
310
345
308.2207
47774.21
350
300
350
316.2278
47434.16
375
250
375
353.5534
44194.17
400
200
400
387.2983
38729.83
425
50
425
418.33
31374.75
450
00
450
447.2136
22360.68
475
50
475
474.3416
1858.54
499
2
499
498.999
498.999
This table shows that the isosceles triangle made of integers with the largest area is very similar to an equilateral triangle. This suggests the largest isosceles triangle is in fact an equilateral triangle.
The Fencing Problem
A farmer has exactly 100 metres of fencing and wants to fence off a plot of land.
She is not concerned about the shape of the plot, but it must have a perimeter of 1000 m.
She wishes to fence off the plot of land, which contains the maximum area.
Investigate the shape, or shapes, that could be used to fence in the maximum area using exactly 1000 m of fencing each time.
Firstly I thought about some of the shapes the possible and decided to calculate their areas in a logical order. So I decided to start with triangles before moving onto quadrilaterals, pentagons, hexagons and so on.
Triangles
To calculate the area of a triangle you must times half the base by the vertical height.
Formulae
A = b x h
Equilateral Triangle
There is more than one way to calculate the area of an equilateral triangle so I will demonstrate each of the three different ways.
. A = x x x sin 60
= 48112.52 (to 2d.p.)
2. A = x - 2 x h
h = sin 60 - - 3
= 48112.52 (to 2d.p.)
3. A = ab sin c
= 48112.52 (to 2d.p.)
Isosceles Triangle
A = b x h
To find the vertical height you must split the isosceles triangle into two separate right-angled triangles and use Pythagoras' Theorem. Pythagoras can only be used in right-angled triangles and uses two sides to find the third side.
h = a2 - (b)sq. = Ans.
h =
Formulae
A = b x
Examples
200 x (90000 - 40000 = 50000.
) = 44721.36 (to 2d.p.)
00 x (160000 - 1000 = 150000
) = 38729.83 (to 2d.p.)
Length of Sides (m)
a
b
c
Height (m)
Area (m )
251
498
251
31.62278
7874.071
275
450
275
58.1139
35575.62
300
400
300
223.6068
44721.36
325
350
325
273.8613
47925.72
330
340
330
282.8427
48083.26
333
334
333
288.0972
48112.23
335
330
335
291.5476
48105.35
340
320
340
300
48000
345
310
345
308.2207
47774.21
350
300
350
316.2278
47434.16
375
250
375
353.5534
44194.17
400
200
400
387.2983
38729.83
425
50
425
418.33
31374.75
450
00
450
447.2136
22360.68
475
50
475
474.3416
1858.54
499
2
499
498.999
498.999
This table shows that the isosceles triangle made of integers with the largest area is very similar to an equilateral triangle. This suggests the largest isosceles triangle is in fact an equilateral triangle.
