= 10.19
Triangle d:
Perimeter = Sin40x80
= 51.42
= 51.42+80+60
= 191.42
Area = s = p ÷ 2
= 191.42 ÷ 2
= 95.71
= square root of s(s-a)(s-b)(s-c)
= square root of 95.71x(95.71-51.42)x(95.71-60)x(95.71-80)
= square root of 95.71x44.29x35.71x15.71
= square root of 2378094.08
= 1542.11
Ratio = a/p
= 1542.11 ÷ 191.42
= 8.06
Triangle e:
Perimeter = 80 ÷ Sin 40
= 124.46
= Sin60x80
= 69.28
= 69.28+124.46+80
=273.74
Area = s = p ÷ 2
= 273.74 ÷ 2
= 136.87
= square root of s(s-a)(s-b)(s-c)
= square root of 136.87x(136.87-80)x(138.67-69.28)x(138.67- 124.46)
= square root of 136.87x56.87x67.59x12.41
= square root of 6528985.79
= 2555.19
Ratio = a/p
= 2555.19 ÷ 273.74
= 9.33
Triangle f:
Perimeter = 60 ÷ Sin40
= 93.34
= Sin80x60
= 59.09
= 93.34+59.09+60
= 212.43
Area = s = p ÷ 2
= 212.43 ÷ 2
= 106.22
= square root of s(s-a)(s-b)(s-c)
= square root of 106.22x(106.22-60)x(106.22-93.34)x(106.22-59.09)
= square root of 106.22x46.22x12.88x47.13
= square root of 2980228.35
= 1726.33
Ratio = a/p
= 1726.33 ÷ 212.43
= 8.13
Triangle g:
Perimeter = 60 ÷ Cos40
= 78.32
= 78.32+80+60
= 218.32
Area = s = p ÷ 2
= 218.32 ÷ 2
= 109.16
= square root of s(s-a)(s-b)(s-c)
= square root of 109.16x(109.16-80)x(109.16-60)x(109.16-78.32)
= square root of 109.16x29.16x30.84
= square root of 98166.98
= 313.32
Ratio = a/p
= 313.32 ÷ 218.32
= 1.44
Triangle h:
Perimeter = Sin40x60
= 38.57
= 38.57+80+60
= 178.57
Area = s = p ÷ 2
= 178.57 ÷ 2
= 89.29
= square root of s(s-a)(s-b)(s-c)
= square root of 89.29x(89.29-67.13)x(89.29-80)x(89.29-60)
= square root of 89.29x22.16x9.29x29.29
= square root of 538403.24
= 733.76
Ratio = a/p
= 733.76 ÷ 178.57
= 4.11
Triangle i:
Perimeter = 40 ÷ Sin60
= 46.19
= Cos60 x 46.19
= 23.1
= 23.1+46.19+40
= 109.29
Area = s = p ÷ 2
= 109.29 ÷ 2
= 54.65
= square root of s(s-a)(s-b)(s-c)
= square root of 54.65x(54.65-40)x(54.65-23.1)x(54.65-46.19)
= square root of 54.65x14.65x31.55x8.46
= square root of 213696.55
= 462.27
Ratio = a/p
= 462.67 ÷ 109.29
= 4.23
Triangle j:
Perimeter = 40 ÷ Cos60
= 80
= 80+80+40
= 200
Area = s = p ÷ 2
= 200 ÷ 2
= 100
= square root of s(s-a)(s-b)(s-c)
= square root of 100x(100-80)x(100-40)x(100-80)
= square root of 100x20x60x20
= square root of 2400000
= 1549.19
Ratio = a/p
= 1549.19 ÷ 200
= 7.75
Triangle k:
Perimeter = Sin60x40
= 34.64
= 34.64+80+40
= 154.64
Area = s = p ÷ 2
= 154.64 ÷ 2
= 77.32
= square root of s(s-a)(s-b)(s-c)
= square root of 77.32x(77.32-34.64)x(77.32-80)x(77.32-40)
= square root of 77.32x42.68x-2.68x37.32
= square root of –330059.84
= impossible to square root a negative
Ratio = a/p
Triangle l:
!Impossible!
Triangle m:
!Impossible!
Triangle n:
Perimeter = Sin80x60
= 59.09
= 59.09+40+60
= 159.09
Area = s = p ÷ 2
= 159.09 ÷ 2
= 79.55
= square root of s(s-a)(s-b)(s-c)
= square root of 79.55x(79.55-59.09)x(79.55-40)x(79.55-60)
= square root of 79.55x20.46x39.55x19.55
= square root of 1258458.98
= 1121.81
Ratio = a/p
= 1121.81 ÷ 159.09
= 7.05
Triangle o:
!Impossible!
Triangle p:
!Impossible!
Triangle q:
Perimeter = Sin60x80
= 69.28
= 69.28+40+80
= 189.28
Area = s = p ÷ 2
= 189.28 ÷ 2
= 94.64
= square root of s(s-a)(s-b)(s-c)
= square root of 94.64x(94.64-69.28)x(94.64-40)x(94.64-80)
= square root of 94.64x25.36x54.64x14.64
= square root of 1919887.36
= 1385.6
Ratio = a/p
= 1385.6 ÷ 189.28
= 7.32
Triangle r:
Perimeter = 40 ÷ Sin80
= 40.62
= Cos60 x 40
= 20
= 40+20+40.62
= 100.62
Area = s = p ÷ 2
= 100.62 ÷ 2
= 50.31
= square root of s(s-a)(s-b)(s-c)
= square root of 50.31x(50.31-40)x(50.31-20)x(50.31-40.62)
= square root of 50.31x10.31x30.31x9.69
= square root of 152343.07
= 390.31
Ratio = a/p
= 390.31 ÷ 100.62
= 3.88
Triangle s:
Perimeter = 40 ÷ Cos80
= 230.35
= 60+40+230.35
= 330.35
Area = s = p ÷ 2
= 330.35 ÷ 2
= 165.18
= square root of s(s-a)(s-b)(s-c)
= square root of 165.18x(165.18-60)x(165.18-40)x(165.18-230.5)
= square root of 165.18x105.18x15.18x-65.32
= square root of –17226957.25
= Impossible to square root a negative
Ratio = a/p
Triangle t:
Perimeter = Sin80x40
= 39.39
= 39.39+60+40
= 139.39
Area = s = p ÷ 2
= 139.39 ÷ 2
= 69.7
= square root of s(s-a)(s-b)(s-c)
= square root of 69.7x(69.7-39.39)x(69.7-60)x(69.7-40)
= square root of 69.7x30.31x1.16x29.7
= square root of 2378094.08
= 1542.11
Ratio = a/p
= 1542.11 ÷ 139.39
= 3.89
Triangle u:
Perimeter = Sin60 x 80
= 69.28
= Cos60 x 80
= 40
= 80+40+69.28
= 189.28
Area = ½ abSinC
= ½ x 69.28 x 80 x Sin 40
= 1781.29
Ratio = a/p
= 1781.29 ÷ 189.28
= 9.41
Triangle v:
!Impossible!
Triangle w:
!Impossible!
Triangle x:
!Impossible!
Triangle y:
!Impossible!
Triangle z:
!Impossible!
Triangle aa:
Perimeter = Sin40x60
= 38.57
= Cos40x60
= 45.96
= 60+45.96+38.57
= 144.53
Area = ½ abSinC
= ½ x 38.57 x 60 x Sin 80
= 1144.84
Ratio = a/p
= 1144.84 ÷ 144.53
= 9.1
Triangle ab:
!Impossible!
Triangle ac:
!Impossible!
Triangle ad:
Perimeter = Sin40x60
= 38.57
= 60+80+38.57
= 178.57
Area = s = p ÷ 2
= 178.57 ÷ 2
= 89.29
= square root of s(s-a)(s-b)(s-c)
= square root of 89.29x(89.29-60)x(89.29-80)x(89.29-38.57)
= square root of 89.29x29.29x9.29x50.72
= square root of 132648.22
= 364.21
Ratio = a/p
= 364.21 ÷ 178.57
= 2.04
Triangle ae:
Perimeter = 40 + 60 + 80
= 180
Area = s = p ÷ 2
= 90
= square root of s(s-a)(s-b)(s-c)
= square root of 90x(90-40)x(90-60)x(90-80)
= square root of 90x50x30x10
= square root of 1350000
= 1161.90
Ratio = a/p
= 1161.90 ÷ 90
= 12.91
Triangle af:
Perimeter = Cos60 x 80
= 40
= 80+40+40
= 160
Area = s = p ÷ 2
= 160 ÷ 2
= 80
= square root of s(s-a)(s-b)(s-c)
= square root of 80x(80-80)x(80-40)x(80-40)
= square root of 80x0x40x40
= square root of 0
= 0
Ratio = a/p
= 0 ÷ 80
= 0
Triangle ag:
Perimeter = 60+80+40
= 180
Area = s = p ÷ 2
= 180 ÷ 2
= 90
= square root of s(s-a)(s-b)(s-c)
= square root of 90x(90-60)x(90-80)x(90-40)
= square root of 90x30x10x10
= square root of 1350000
= 1161.9
Ratio = a/p
= 1161.9 ÷ 180
= 6.45
Triangle ah:
Perimeter = Sin80 x 40
= 39.39
= 40+60+39.39
= 139.39
Area = s = p ÷ 2
= 139.39 ÷ 2
= 69.7
= square root of s(s-a)(s-b)(s-c)
= square root of 69.7x(69.7-40)x(69.7-60)x(69.7-39.39)
= square root of 69.7x29.7x9.7x30.31
= square root of 608620.95
= 780.14
Ratio = a/p
= 780.14 ÷ 139.39
= 5.58
Triangle ai:
Perimeter = 80+40+60
= 180
Area = s = p ÷ 2
= 180 ÷ 2
= 90
= square root of s(s-a)(s-b)(s-c)
= square root of 90x(90-80)x(90-40)x(90-60)
= square root of 90x10x50x30
= square root of 1350000
= 1161.9
Ratio = a/p
= 1161.9 ÷ 180
= 6.45
I will now try to find the ratio of the triangle “a”. The ratio is impossible to find using the formulas that are mention at the beginning. The only way is to draw the triangle and measure the sides from that drawing. I will do this below:
(Scale = 1:2)
The measurements are:
a = 43.6cm
b = 50
c = 31cm
Perimeter = 43.6 + 50 + 31
= 124.6cm
Area = s = p ÷ 2
= 124.6 ÷ 2
= 62.3
= square root of s(s-a)(s-b)(s-c)
= square root of 62.3x(62.3-43.6)x(62.3-50)x(62.3-31)
= square root of 62.3x18.7x12.3x31.3
= square root of 448517.2
= 669.71
Ratio = a/p
= 669.71 ÷ 124.6
= 5.37
Below is a table to show the results of the ratios of the triangles that I found. The table will show any triangles that are the same or anomalous:
I will now draw a graph to show the above results in a more graphical way. This will allow me to see if any triangles are the same by looking at the ratios:
I have found that 13 triangles are either impossible to draw or are impossible to calculate. These are k, l, m, o, p, s, v, w, x, y, z, ab and ac. I have found that two pairs of triangles are the same. These are b & ae and ag & ai.
The largest ratio I have found, using the numbers 40, 60 and 80 are the triangles b and ae. This is one pair of triangles that are the same. Below is a diagram of triangle b with the numbers 40, 60 and 80:
40 80
60
To extend this piece of work further, I would like to try using numbers that follow the same pattern as the numbers 40, 60 and 80. I would choose numbers like 10, 30 & 50 and 20, 40 & 60 etc. Then I would increase the gap between the numbers and use numbers like 10, 40 & 70, and 30, 60 & 90. Also, instead of changing numbers, I could change the shapes using such shapes as square, pentagon, hexagon, etc.