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    In my coursework i have got 7 formulas, as shown below:

    triangular: ½ n2 + ½ n
    pentagonal: (2.5n2 – 2.5n) + 1
    hexagonal: (3n2 – 3n) + 1
    heptagonal: (3.5n2 – 3.5n) + 1
    octagonal: (4n2 – 4n) + 1
    nonaonal: (4.5n2 – 4.5n) + 1
    decagonal: (5n2 – 5n) + 1

    it then tells me to derive a general equation which links to all polygonal bases, making each point of how i obtained the formula clearly explained (by explaining the stages of the process), in detail.

    personally i dont know how to do this, so i was wondering if you could help me.


    You can split up the k-sided polygon into the dot in the middle, plus the k "spokes" going from the centre to the vertices, plus the k triangles between the spokes. (See the attached picture.) Each spoke contains (n - 1) dots. Each triangle contains

    (n - 2) + (n - 3) + ... + 3 + 2 + 1 = (1/2)(n - 1)(n - 2)

    dots. So the total number of dots is

    1 + k(n - 1) + (1/2)k(n - 1)(n - 2)
    = 1 + k(n - 1) * [1 + (1/2)(n - 2)]
    = 1 + (1/2)kn(n - 1).
    Attached Images
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Updated: July 22, 2004
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