A table to show the area of a rectangle with a perimeter of 1000m made up of different combinations of lengths of its sides
At a glance of the table, I will be able to distinguish which combination gives the maximum area.
When this is distinguished for all the shapes, I will show the maximum areas for each shape on a separate table as follows.
A table to show the maximum area of a number of shapes found using a number of combinations of lengths of the sides
Finally I will display my results in the form of a bar graph as with this type of graph it is easy to see at a glance the areas of each of the shapes and which shape will have the greatest area. I will also draw a line graph which will show how the areas of the shapes get closer to the circle (both have the same axis).
A graph to show the maximum area of a number of shapes found using a number of different lengths of the sides.
Triangle
The first group of shapes that I am going to investigate is the triangles.
Firstly there is only on area for the equilateral triangle. As this type of triangle has all equal sides and angles. Therefore it is impossible for this shape to have a different area.
Height a²=b²+c²
333⅓²=16.67²+c²
c²= 333⅓²- 166.67²
c²=83 333⅓
c= 288.68 metres
Area = ½bh
=½ x333⅓ x 288.68
=48 122.52 metre²
OR
Area = ½bc SinA
=½ x 333⅓ x 333⅓ x sin 60º
=48 122.52
For the isosceles triangle I will take an example of different bases and heights to find out which combination gives the maximum area.
Number 1
a=100 b=c=450
height a²=b²+c²
50²=450²+c²
c²=450²-50²
c²= 200 00
c+ 447.21 meters
Area=½bh
=½x 100x 447.21
= 22 360.68 metres²
Number 2
a=200 b=c=400
height a²=b²+c²
100²=400²+c²
c²=400²-100²
c²=150 000
c=387.29 meters
Area=½bh
=½x200x387.29
=38729.83 metres²
Number 3
a=300 b=c=350
height a²=b²+c²
150²=350²+c²
c²=350²-150²
c²=100 000
c=316.23 meters
Area=½bh
=½x 300x316.23
=47 434.16 metres²
Number 4
a=400 b=c=300
height a²=b²+c²
200²=300²+c²
c²=300²-200²
c²=50 000
c=223.61 meters
Area=½bh
=½x400x223.61
=44 722 metres²
The isosceles triangle which has the maximum area out of the ones which I have investigated is number 3. Number 3 has a base of 300 meters and a height of 316.23 meters. It also has an area of 47 434.16 metres²
I am now going to investigate a right angle triangle.
Ratio of length of sides
a : b : c
3 : 4 : 5
1=1000= 83½m
12
a = 3 x 83½= 250m
b = 4 x 83½= 333⅓m
c = 5 x 83½= 416⅔m
Angle A
a = c
SinA SinC
250 = 416⅔
SinA = Sin 90º
SinA x 416⅔= Sin 90º x 250
Sin A = Sin90º x 250
416⅔
SinA = 0.6
A=36.87º
Area= ½bc SinA
=½x333⅓x416⅔x Sin36.87º
=41 666.77 m²
The last type of triangle I am going to invest is the scalene.
Number 1
a=400, b= 450 & c=150
CosA= b²+c²-a²
2bc
CosA=450²+150²-400²
2x450x150
CosA= 65 000
135 000
CosA=0.48
A=61.22º
Area= ½bcSinA
=½x450x150xSin61.22º
=29 581.02
Number 2
a=350 b=450 c=200
CosA=b²+c²-a²
2bc
CosA=450²+200²-350²
2x450x200
CosA=120 000
180 000
CosA=0.67
A=48.19º
Area= ½bcSinA
=½x450x200xSin48.19º
=33 541.18m²
