Today, I noticed a pattern in the difference between interior angles of regular polygons. This led to an investigation on Sheets to try and find the pattern. Finally, we tried to figure out if the pattern is exponential or linear. The pattern is. (Sorry. Almost forgot that the conclusion comes after the results!)
| Number of Sides | Interior Angle(°) | Difference to Previous Interior Angle (°) |
| 3 | 60.000 | 0.000 |
| 4 | 90.000 | 30.000 |
| 5 | 108.000 | 18.000 |
| 6 | 120.000 | 12.000 |
| 7 | 128.571 | 8.571 |
| 8 | 135.000 | 6.429 |
| 9 | 140.000 | 5.000 |
| 10 | 144.000 | 4.000 |
| 11 | 147.273 | 3.273 |
| 12 | 150.000 | 2.727 |
| 13 | 152.308 | 2.308 |
| 14 | 154.286 | 1.978 |
| 15 | 156.000 | 1.714 |
| 16 | 157.500 | 1.500 |
| 17 | 158.824 | 1.324 |
| 18 | 160.000 | 1.176 |
| 19 | 161.053 | 1.053 |
| 20 | 162.000 | 0.947 |
| 21 | 162.857 | 0.857 |
| 22 | 163.636 | 0.779 |
| 23 | 164.348 | 0.711 |
| 24 | 165.000 | 0.652 |
| 25 | 165.600 | 0.600 |
| 26 | 166.154 | 0.554 |
| 27 | 166.667 | 0.513 |
| 28 | 167.143 | 0.476 |
| 29 | 167.586 | 0.443 |
| 30 | 168.000 | 0.414 |
| 31 | 168.387 | 0.387 |
| 32 | 168.750 | 0.363 |
| 33 | 169.091 | 0.341 |
| 34 | 169.412 | 0.321 |
| 35 | 169.714 | 0.303 |
| 36 | 170.000 | 0.286 |
| 37 | 170.270 | 0.270 |
| 38 | 170.526 | 0.256 |
| 39 | 170.769 | 0.243 |
| 40 | 171.000 | 0.231 |
| 41 | 171.220 | 0.220 |
| 42 | 171.429 | 0.209 |
| 43 | 171.628 | 0.199 |
| 44 | 171.818 | 0.190 |
| 45 | 172.000 | 0.182 |
| 46 | 172.174 | 0.174 |
| 47 | 172.340 | 0.167 |
| 48 | 172.500 | 0.160 |
| 49 | 172.653 | 0.153 |
| 50 | 172.800 | 0.147 |
| 51 | 172.941 | 0.141 |
| 52 | 173.077 | 0.136 |
| 53 | 173.208 | 0.131 |
| 54 | 173.333 | 0.126 |
| 55 | 173.455 | 0.121 |
| 56 | 173.571 | 0.117 |
| 57 | 173.684 | 0.113 |
| 58 | 173.793 | 0.109 |
| 59 | 173.898 | 0.105 |
| 60 | 174.000 | 0.102 |
| 61 | 174.098 | 0.098 |
| 62 | 174.194 | 0.095 |
| 63 | 174.286 | 0.092 |
| 64 | 174.375 | 0.089 |
| 65 | 174.462 | 0.087 |
| 66 | 174.545 | 0.084 |
| 67 | 174.627 | 0.081 |
| 68 | 174.706 | 0.079 |
| 69 | 174.783 | 0.077 |
| 70 | 174.857 | 0.075 |
| 71 | 174.930 | 0.072 |
| 72 | 175.000 | 0.070 |
| 73 | 175.068 | 0.068 |
| 74 | 175.135 | 0.067 |
| 75 | 175.200 | 0.065 |
| 76 | 175.263 | 0.063 |
| 77 | 175.325 | 0.062 |
| 78 | 175.385 | 0.060 |
| 79 | 175.443 | 0.058 |
| 80 | 175.500 | 0.057 |
| 81 | 175.556 | 0.056 |
| 82 | 175.610 | 0.054 |
| 83 | 175.663 | 0.053 |
| 84 | 175.714 | 0.052 |
| 85 | 175.765 | 0.050 |
| 86 | 175.814 | 0.049 |
| 87 | 175.862 | 0.048 |
| 88 | 175.909 | 0.047 |
| 89 | 175.955 | 0.046 |
| 90 | 176.000 | 0.045 |
| 91 | 176.044 | 0.044 |
| 92 | 176.087 | 0.043 |
| 93 | 176.129 | 0.042 |
| 94 | 176.170 | 0.041 |
| 95 | 176.211 | 0.040 |
| 96 | 176.250 | 0.039 |
| 97 | 176.289 | 0.039 |
| 98 | 176.327 | 0.038 |
| 99 | 176.364 | 0.037 |
| 100 | 176.400 | 0.036 |
As the number of sides increase, the interior angle increases. The difference between the interior angles decreases inversely to the number of sides. As the interior angles grow, the difference between them gets smaller. By using these statements and the results, we created a formula for finding the difference between interior angles (x) with the number of sides (n):
x=360÷(n2+n)
The pattern involving the number of sides and difference between interior angles is quadratic. This is due to the highest exponent (power) being 2. Specifically, this pattern is an example of quadratic decay because the difference decreases inversely with the square of the number of sides.