The Fencing Problem.

175000=b²

418.3300133=b

area= 418.3300133 x 75

=31374.751m²

= 31374.8m² (1dp)

c ²=a²+b²

375²=125² +b²

140625 - 15625=b²

125000= b²

353.5533906=b

area= 353.5533906 x 125

= 44194.17382m²

= 44194.2m²(1dp)

c ²=a²+b²

350²= 150²+b²

122500 - 22500=b²

100000=b²

316.227766=b

area=316.227766 x 150

=47925.72378m²

= 47925.7m²(1dp)

c ²=a²+b²

325² =175² +b² 105625-30625=b²

75000=b²

273.8612788=b

area=273.8612788 x 175 = 47925.72378m²

= 47925.7m² (1dp)

c ²=a²+b²

300²=200²+b²

90000-40000=b²

50000=b²

223.6067978=b

area= 223.6067978 x 200

= 44721.35955m²

= 44721.4m² (1dp)

c ²=a²+b²

275²=225²+b²

75625-50625=b²

25000=b²

158.113883=b

area= 158.113883 x 225

= 35575.62368m²

= 3557.6m² (1dp)

c ²=a²+b²

334²=166²+b²

111556 - 27556=b² 84000=b²

289.8275349=b

area= 289.8275349 x 166

= 4811.3708m²

= 4811.4m² (1dp)

The triangle with the biggest area is the equilateral triangle and so far this shows the shape with the equal sides has the biggest area like the square. To show this I have plotted my results on a graph and on a table.

Regular Polygons:

Regular polygons have the same length on each side and the same interior and exterior angle.

Pentagon:

A pentagon has 5 sides.

l=1000=200

5

To find the area of a pentagon you can divide it into 5 triangles. I found the area of 1 and

multiplied it by 5. To find the area of the

triangle I used trigonometry. To find the angle I found the external angle first by doing:

360= 72= external angle

5

180-72=108= interior angle (divided the interior angle by 2 to get the angle of the triangle)

Tangent=opposite adjacent

Tan 54° = x

100

= 137.638

= 137.6 (1dp)

Area of triangle= 1/2base x height

= 100 x 137.6

= 1376.38

Area of pentagon = 1376.38 x 5

= 68819m²

Hexagon: A hexagon has 6 sides.

l=1000 = 166.666

6

To find the area of a hexagon you can divide

it into 6 triangles. I found the area of 1 and

multiplied it by 6. To find the area of a triangle I used triganometry. To find the angle I found the external angle first by doing:

360= 60= external angle

6

180-60=120= interior angle(divided the interior angle by 2 to get the angle of the triangle)

Tangent=opposite

adjacent

Tan 60°= x

83.33333

= 144.3375673

Area of triangle=1/2base x height

=144.3375673 x 83.33333

=12028.13056

Area of hexagon =12028.13056 x 6

= 72168.78336

= 72168.8m² (1dp)

Heptagon: A heptagon has 7 sides.

l=1000 = 142.85714429

7

To find the area of a heptagon you can divide it into 7 triangles. I found the area of 1 and multiplied it by 7. To find the area of a triangle I used triganometry. 360= 51.43= external angle

7

180-51.43=128.57= interior angle(divided the interior angle by 2 to get the angle of the triangle)

Tangent=opposite

adjecent

Tan 66.285° = x = 148.3182269

71.428

Area of triangle=1/2base x height

= 71.428 x 148.3182269

= 10594.07431 x 7

Area of hexagon = 71458.52018

= 71458.5m² (1dp)

Octagon An octagon has 8 sides.

l=1000= 125

8

To find the area of a octagon you can divide it into 8 triangles. I found the area of 1 and multiplied it by 8. To find the area of a triangle I used triganometry.

360= 45= external angle

8

180-45=135= interior angle(divided the interior angle by 2 to get the angle of the triangle)

Tangent=opposite

adjecent

Tan 67.5° = x = 150.8883476

62.5

Area of triangle=1/2base x height

=62.5 x 150.8883476

= 9430.521728

Area of octagon = 9430.521728 x 8

= 75444.17382

= 75444.2m² (1dp)

Nonagon: An nonagon has 9 sides

l=1000= 111.1111

9

To find the area of an nonagon you can divide it into 9 triangles. I found the area of 1 and multiplied it by 9. To find the area of a triangle I used triganometry. 360= 40= external angle

9

180-40=140= interior angle(divided the interior angle by 2 to get the angle of the triangle)

Tangent=opposite

adjecent

Tan 20°= x = 152.6374818

55.5555

Area of triangle=1/2base x height

= 55.5555 x 152.6374818

= 8479.851619

Area of nonagon = 8479.851619 x 9

= 76318.66457

= 76318.7m² (1dp)

Decagon: A decagon has 10 sides.

l= 1000= 100

10

To find the area of a decagon you can divide it into 10 triangles. I found the area of 1 and multiplied it by 10. To find the area of a triangle I used triganometry. 360= 36= external angle

10

180-36=144= interior angle (divided the interior angle by 2 to get the angle of the triangle)

Tangent=opposite

adjecent

Tan 72°= x

50

=153.8

Area of triangle=1/2base x height

= 50 x 153.8

= 7694.208843

Area of decagon= 7694.208843 x 10

= 76942.08843

= 76942.1m² (1dp)

Here is a table to show my results and also a graph to show my results of the polygon shapes.

Circle: The circle can only have one radius if the curcumference is 1000m.

a=??r²

p= ??d

1000=??d

1000= 318.309

?????????????????????d

r= 318.309

2

area= 159.1549431² x ?

???????????????????????????

???????????????????????m² (1dp)

Here is a table to show all my results:

Formula: By finding out the area of the shapes we can see that the circle has the largest area so far and that no polygon/shape can have a bigger area. We can put this in a fromula and to prove that no shape has the area bigger than a circle as we could replace n with 1000 and still the area would be smaller than the circle. Here is how i worked out the formula.

Length of side= 1000

n

Exterior angle= 360

n

Interior angle= 180 - 360

n

1/2 interior angle= 90 - 180

n

Height of triangle= 1/2 length side x tan 1/2interior angle

Height = 500 tan(90 - 180)

n n

Area of triangle= 1/2base x height = 1/2 x 1000 x 500 tan (90 - 180)

n n n

Area of polygon= n x 1/2 x 1000 x 500 tan (90 - 180)

n n n

I then tried it to make sure it worked by replacing n with number of sides which i had tried before. I tried it with my first regular polygon a pentagon.

Area of pentagon= 5 x 1/2 x 1000 x 500 tan (90 - 180)

5 5 5

= 5 x1/2 200 x 100 x tan 54= 68819 m²

I then tried to find the area of polygons with more sides using my formula to see if the area was bigger than my other shapes.

Area of shape with 100 sides= 100 x 1/2 x 1000 x 500 tan (90 - 180)

100 100 100

= 100 x 1/2 10 x 5 x tan 88.2= 79551.28988

= 79551.3m² 1dp

Area of shape with 200 sides= 200 x 1/2 x 1000 x 500 tan (90 - 180)

200 200 200

= 200 x 1/2 x 5 x 2.5x tan 89.1= 79570.92645

= 79570.9m² 1dp

Area of shape with 300 sides= 300 x 1/2 x 1000 x 500 tan (90 - 180)

300 300 300

= 300 x 1/2 x 3.33333 x 1.666x tan 89.4= 7957137.886

= 79570.6m² 1dp

Area of shape with 400 sides=400 x 1/2 x 1000 x 500 tan (90 - 180)

400 400 400

= 400 x 1/2 x 2.5 x 1.25 x tan 89.55= 79576.42435

= 79575.8m² 1dp

Area of shape with 500 sides= 500 x 1/2 x 1000 x 500 tan (90 - 180)

500 500 500

= 500 x 1/2 2 x 1 x tan 89.64= 79576.42435

= 79576.4m² 1dp

Area of shape with 600 sides=600 x 1/2 x 1000 x 500 tan (90 - 180)

600 600 600

= 600 x 1/2 1.666 x 0.833x tan 89.7= 79576.50771

= 79576.5m² 1dp

Area of shape with 700 sides=700 x 1/2 x 1000 x 500 tan (90 - 180)

700 700 700

= 700 x 1/2 1.428571429 x 0.714285714 x tan 89.74285714

= 79576.93727

= 79576.9m² 1dp

Area of shape with 800 sides= 800 x 1/2 x 1000 x 500 tan (90 - 180)

800 800 800

= 800 x 1/2 1.25 x 0.625 x tan 89.775= 79577.06248

= 79577.1m² 1dp

Area of shape with 900 sides= 900 x 1/2 x 1000 x 500 tan (90 - 180)

900 900 900

= 900 x 1/2 1.1111 x 0.5555 x tan 89.8= 79577.14834

= 79577.1m² 1dp

Area of shape with 1000 sides= 1000 x 1/2 x 1000 x 500 tan (90 - 180)

1000 1000 1000

= 1000 x 1/2 1 x 0.5 x tan 89.82= 79577.2097

= 79576.2m² 1dp

Conclusion:

The circle has the biggest area with the perimeter 1000m. This is because the shape with the most equal sides has the biggest area. This was shown first in the square and then the equalteral triangle. I then tried to see if this worked on all the shapes by using regular polygons. This proved that I was right as the decagon had a bigger area than the equalateral triangle. The circle has inmfinate number of sides which are all equal. The more sides you had the bigger the surface area got. From using my formula you can see that no shape has an area bigger than the circle not even a shape with 1000 sides. The area only went up slightly from 100 sides to 1000 and this showed that the area only increased slightly. This shows that the circle would be the best shape for the farmer to plot her land.