Tan 54 = h
100
1.376 x 100 = h
138 = h
½ x 200 x 138 = 13800
13800 x 5 = 69000
Area = 69000m²
Hexagons
I am now going to look at a hexagon to find out its area when using 1000meters of fencing.
Formula To Find the Area Of A Hexagon:
Exterior Angle = 360°
No. Of Sides
Interior Angle = 180°-Exterior Angle
I will then use tangent of an angle, to work out the area of one of the six segments of the hexagon. I will multiply this number by 6 because the 6 segments are all the same size. Therefore I will get the overall area of the hexagon.
Hexagon Area
Area: Ex. Angle = 360° = 60°
6
Int. Angle = 180°-60°=120°
= 120°÷2=60°
Tan = opp
adj
Tan 60 = h
100
1.73 x 83.33 = h
144.2 = h
½ x 166.66 x 144.2 = 12016.2
12016.2 x 6 = 72097.2
Area = 72097.2m²
Heptagons
I am now going to look at a heptagon to find out its area when using 1000meters of fencing.
Formula To Find the Area Of A Heptagon:
Exterior Angle = 360°
No. Of Sides
Interior Angle = 180°-Exterior Angle
I will then use tangent of an angle, to work out the area of one of the seven segments of the heptagon. I will multiply this number by 7 because the 7 segments are all the same size. Therefore I will get the overall area of the heptagon.
Heptagon Area
Area: Ex. Angle = 360° = 51° (51.43°)
7
Int. Angle = 180°-51.43°=128.6°
= 128.6°÷2=64.3°
Tan = opp
adj
Tan 64.3 = h
71.4275
2.08 x 71.4275 = h
148.6 = h
½ x 142.855 x 148.6 = 10614.1
10614.1 x 7 = 74298.7
Area = 74298.7m²
Octagons
I am now going to look at an Octagon to find out its area when using 1000meters of fencing.
Formula To Find the Area Of A Octagon:
Exterior Angle = 360°
No. Of Sides
Interior Angle = 180°-Exterior Angle
I will then use tangent of an angle, to work out the area of one of the eight segments of the octagon. I will multiply this number by 8 because the 8 segments are all the same size. Therefore I will get the overall area of the octagon.
Octagon Area
Area: Ex. Angle = 360° = 45°
8
Int. Angle = 180°-45°=135°
= 135°÷2=67.5°
Tan = opp
adj
Tan 67.5 = h
62.5
2.41 x 62.5 = h
150.625 = h
½ x 125 x 150.625 = 9414.0625
9414.0625 x 8 = 75312.5
Area = 75312.5m²
Nonagons
I am now going to look at a Nonagon to find out its area when using 1000meters of fencing.
Formula To Find the Area Of A Nonagon:
Exterior Angle = 360°
No. Of Sides
Interior Angle = 180°-Exterior Angle
I will then use tangent of an angle, to work out the area of one of the nine segments of the nonagon. I will multiply this number by 9 because the 9 segments are all the same size. Therefore I will get the overall area of the nonagon.
Nonagon Area
Area: Ex. Angle = 360° = 40°
9
Int. Angle = 180°-40°=140°
= 140°÷2=70°
Tan = opp
adj
Tan 70 = h
55.55
2.75 x 55.55 = h
152.8(1d.p) = h
½ x 111.1 x 152.8 = 8488.04
8488.04 x 9 = 76392.36
Area = 76392.36m²
Decagons
I am now going to look at a Decagon to find out its area when using 1000meters of fencing.
Formula To Find the Area Of A Decagon:
Exterior Angle = 360°
No. Of Sides
Interior Angle = 180°-Exterior Angle
I will then use tangent of an angle, to work out the area of one of the ten segments of the decagon. I will multiply this number by 10 because the 10 segments are all the same size. Therefore I will get the overall area of the decagon.
Decagon Area
Area: Ex. Angle = 360° = 36°
10
Int. Angle = 180°-36°=144°
= 144°÷2=72°
Tan = opp
adj
Tan 72 = h
50
3.078 x 50 = h
153.9 = h
½ x 100 x 153.9 = 7695
7695 x 10 = 76950
Area = 76950m²
Dodecagon Shape
I am now going to look at a 12-sided Shape to find out its area when using 1000meters of fencing.
Formula To Find the Area Of A Dodecagon:
Exterior Angle = 360°
No. Of Sides
Interior Angle = 180°-Exterior Angle
I will then use tangent of an angle, to work out the area of one of the twelve segments of the dodecagon. I will multiply this number by 12 because the 12 segments are all the same size. Therefore I will get the overall area of the 12-sided Shape.
Area: Ex. Angle = 360° = 30°
12
Int. Angle = 180°-30°=150°
= 150°÷2=75°
Tan = opp
adj
Tan 70 = h
41.67
3.732 x 41.67 = h
155.5 = h
½ x 83.33 x 155.5 = 6478.91
6478.91 x 12 = 77792.22
Area = 77746.92m²
20 Sided Shape
I am now going to look at a 20-sided Shape to find out its area when using 1000meters of fencing.
Formula To Find the Area Of A 20-sided shape:
Exterior Angle = 360°
No. Of Sides
Interior Angle = 180°-Exterior Angle
I will then use tangent of an angle, to work out the area of one of the twenty segments of the shape. I will multiply this number by 20 because the 20 segments are all the same size. Therefore I will get the overall area of the 20-sided Shape.
Area: Ex. Angle = 360° = 18°
20
Int. Angle = 180°-18°=162°
= 162°÷2=81°
Tan = opp
adj
Tan 81 = h
25
6.31375 x 25 = h
157.84 = h
½ x 50 x 157.84 = 3946
3946 x 20 = 78920
Area = 78920m²
40-Sided Shape
I am now going to look at a 40-sided Shape to find out its area when using 1000meters of fencing.
Formula To Find the Area Of A 40 Sided Shape:
Exterior Angle = 360°
No. Of Sides
Interior Angle = 180°-Exterior Angle
I will then use tangent of an angle, to work out the area of one of the forty segments of the shape. I will multiply this number by 40 because the 40 segments are all the same size. Therefore I will get the overall area of the 40-sided Shape.
Area: Ex. Angle = 360° = 9°
40
Int. Angle = 180°-9°=171°
= 171°÷2=85.5°
Tan = opp
adj
Tan 85.50 = h
12.5
12.7062 x 12.5 = h
158.8275 = h
½ x 25 x 158.8275 = 1985.334
1985.334 x 40 = 79413.76
Area = 79413.76m²
60-Sided Shape
I am now going to look at a 60-sided Shape to find out its area when using 1000meters of fencing.
Formula To Find the Area Of A 60 Sided Shape:
Exterior Angle = 360°
No. Of Sides
Interior Angle = 180°-Exterior Angle
I will then use tangent of an angle, to work out the area of one of the sixty segments of the shape. I will multiply this number by 60 because the 60 segments are all the same size. Therefore I will get the overall area of the 60-sided Shape.
Area: Ex. Angle = 360° = 6°
60
Int. Angle = 180°-6°=174°
= 174°÷2=87°
Tan = opp
adj
Tan 87 = h
8.334
19.08114 x 8.334 = h
159.0222 = h
½ x 16.667 x 159.0222 = 1324.6549
1324.6549 x 60 =
Area = 79479.29m²
80-Sided Shape
I am now going to look at an 80-sided Shape to find out its area when using 1000meters of fencing.
Formula To Find the Area Of A 80 Sided Shape:
Exterior Angle = 360°
No. Of Sides
Interior Angle = 180°-Exterior Angle
I will then use tangent of an angle, to work out the area of one of the eighty segments of the shape. I will multiply this number by 80 because the 80 segments are all the same size. Therefore I will get the overall area of the 80-sided Shape.
Area: Ex. Angle = 360° = 4.5°
80
Int. Angle = 180°-4.5°=175.5°
= 175.5°÷2=87.75°
Tan = opp
adj
Tan 87.75 = h
6.25
25.4517 x 6.25 = h
159.073 = h
½ x 12.5 x 159.073 = 994.2063
994.20 x 80 =
Area = 79536m²
100-Sided Shape
I am now going to look at a 100-sided Shape to find out its area when using 1000meters of fencing.
Formula To Find the Area Of A 100 Sided Shape:
Exterior Angle = 360°
No. Of Sides
Interior Angle = 180°-Exterior Angle
I will then use tangent of an angle, to work out the area of one of the one hundred segments of the shape. I will multiply this number by 100 because the 100 segments are all the same size. Therefore I will get the overall area of the 100-sided Shape.
Area: Ex. Angle = 360° = 3.6°
100
Int. Angle = 180°-3.6°=176.4°
= 176.4°÷2=88.2°
Tan = opp
adj
Tan 88.2 = h
5
31.8205 x 5 = h
159.1025 = h
½ x 10 x 159.1025 = 795.5125
795.5125 x 100 = 79551.25
Area = 79551.25m²
250-Sided Shape
I am now going to look at a 250-sided Shape to find out its area when using 1000meters of fencing.
Formula To Find the Area Of A 250 Sided Shape:
Exterior Angle = 360°
No. Of Sides
Interior Angle = 180°-Exterior Angle
I will then use tangent of an angle, to work out the area of one of the two hundred and fifty segments of the shape. I will multiply this number by 250 because the 250 segments are all the same size. Therefore I will get the overall area of the 250-sided Shape.
Area: Ex. Angle = 360° = 1.44°
250
Int. Angle = 180°-1.44°=178.56°
= 178.56°÷2=89.28°
Tan = opp
adj
Tan 89. 28 = h
2
79.5733 x 2 = h
159.1466 = h
½ x 4 x 159.1466 = 318.2932
318.2932 x 250 =
Area = 79573.3m²
500-Sided Shape
I am now going to look at a 500-sided Shape to find out its area when using 1000meters of fencing.
Formula To Find the Area Of A 500 Sided Shape:
Exterior Angle = 360°
No. Of Sides
Interior Angle = 180°-Exterior Angle
I will then use tangent of an angle, to work out the area of one of the five hundred segments of the shape. I will multiply this number by 500 because the 500 segments are all the same size. Therefore I will get the overall area of the 500-sided Shape.
Area: Ex. Angle = 360° = 0.72°
500
Int. Angle = 180°-0.72°=179.28°
= 179.28°÷2=89.64°
Tan = opp
adj
Tan 89.64 = h
1
159.1529 x 1 = h
159. 1529 = h
½ x 2 x 159. 1529 = 159.1529
159.1529 x 500 =
Area = 79576.45m²
Circle
I am now going to work out the area of a circle when using 1000 meters of fencing.
Formula To Find The Area Of A Circle: πr²
First I will find the diameter of the circle, using the circumference formula, which I will then half the diameter to get the radius, therefore I will then be able to work out the area of the circle.
Area Of A Circle
Circumference = π x Diameter
1000 = π x D
1000 = D
π
318.31m=Diameter (Radius=159.155m)
Area = π x r²
= π x 159.155²
= π x 25330.31
=79577.52m²
Table Of Results
I have put the area of all the shapes I have investigated into a table to compare their area and this will give you an idea about which shape has the biggest area.
Conclusion
I was set the task of finding a shape with the biggest area when using 1000meters of fencing, so that a farmer could fence a plot of her land. The shape didn’t matter, however it had to have the maximum area.
I investigated regular shapes because it is only when all the sides are equal I will get the highest area of that shape.
From investigating different shapes I overall found that a circle had the maximum area. A circles area when using 1000meters of fencing is 79577.52m².
I noticed that the area didn’t change much when I started investigating the area of a Nine sided shapes and onwards. Also when I reached a 40-sided shape the area change in a very small amount. As you can see from my graph (on the previous page) the line goes up then curves round and becomes a constant line until we reach the area of a circle, where it stops as I found out because a circle has the maximum area.
In conclusion I have found that the more sides a shape has the bigger the area will be until you get to a circle, which as I found out has the maximum area.