The Fencing Problem.
A farmer has exactly 1000 metres of fencing and wants to fence off a plot of level land. She is not concerned about the shape of the plot, but it must have a perimeter of 1000m. Which shape, with a perimeter of 1000m has the maximum possible area?
Let us start off with a isosceles triangles. The area of an isosceles triangle can be computed by using the Area= ½ base x height formula. Let us start of with the equilateral triangle.
Then each side has length 1000/3=333.4. We need to calculate the height. For this we use trigonometry. The height(h) divides the triangle in two identical right angles. All the angles are equal and add up to 180° so angle C is 60°. Then, since,
tan C= height/adjacentwe have that height=tanC x adjacent
So area of triangle is ½ x 166.67 x 333.34 x tan 60 = 48114.4
Next suppose we shorten the base so that it’s 300 m in length. Then the other two sides are 350 m each. Here we don’t know the angles but we can use Pythagoras’ theorem to obtain the height. We get 150^2+h^2=350^2 which means h^2= 350^2-150^2=100000 and h=316.2So area of triangle is ½ x 300 x 316.2=47434.2
Next we shorten the base consecutively another four times and perform the same calculation. The results are displayed in the table below.
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Base m
Side m
Area m^2
334.34
333.34
48114.4
300
350
47434.2
250
375
44194.2
200
400
38729.8
150
425
31374.8
100
450
22360.7
It is clear from the table that the area decreases and that the equilateral triangle has the largest area.
Next we consider rectangles. A square with a perimeter of 1000 m has sides of length 250 m and thus the area 250×250=62 500 squared meters.
Let us stretch the square by 25 meters to 275. To keep the 1000 m perimeter the horizontal sides shorten to 225. The area is 275×225=61875.We repeat this procedure and show the results in the table below.
Hight m
Width m
Area m^2
250
250
62000
275
225
61875
300
200
60000
325
175
56875
350
150
52500
375
125
46875
We can see that the squre has the greatest area and that the area decrases as the square is stretched. We can see that the area would eventually be zero as the wiidth gets smaller and smaller and close to zero.
Also we see that the the square has a greater area than the triangle. Let us examine shapes with more sides to see if the area increases.
Thus we should calculate the agrea of a regular pentagon. It can be divided into 5 isosceles triangles with each of the sides as a base. Using the same procedure as when we calculated the area of the equilateral triangle we can calculte the area of the triangle in the pentagon. Multiplying by 5 we get the area of the pentagon.
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Each side has length 1000/5= 200. The angle at the centre is 360/5=72. As this is an isosceles triangle we get that the other two angles are (180-72)/2=54. So height=tan54 x 100 and the area of the triangle is ½ 200 x tan 54 x 100=13763.8. Now we need to multiply by 5 to get the area of the triangle. We get Area= 68819.1 m
This is bigger than the square.
Let us construct a formula for the general n-sided polygon. The length od each side is 1000/n. The central angle is 360/n. So the other two angles are (180-360/n)/2 which simplifies to 90 –180/n. That gives the formula for the height h=tan(90 –180/n) x (1000/2n)and the area of triangle is ½ 1000/n x tan(90 –180/n) x (1000/2n).We need to multiply the number of sides which givesArea=500 x tan(90 –180/n) x 500/n.
Using this formula we calculate areas for a n sided polygon as n increases. The results are shown in the table below.
Number of sides
Side length
Area
5
200
68819.1
6
166.7
72168.8
7
142.9
74161.5
8
125
75444.2
9
111.1
76318.8
10
100
76942.1
11
90.1
77401.9
15
66
78410.5
30
33.3
79286.4
1000
1
79577.2
It is clear that as the number of sides increases so does the area. But as the number of sides increases we get closer and closer to the shape of a circle which can be thought of as the an infinitely sided polygon. Thus the shape with the largest area is the ciscle.
What is the area of a circle with perimeter of 1000m? 2x Pi x radius=1000 so radius=159.2. From the formula for the area of a circle, Pi x r^2, we get the area 79622.53
