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It is an octagon with 8 sides.

Let n be the number of sides = number of vertices.

Then each vertex can be connected to n-3 other vertices, giving n(n -3) connections. However, this counts each vertex pair connection twice, so there are:

n(n - 3) / 2 diagonals.

Thus if there are 20 diagonals:

n(n - 3) / 2 = 20

n2 - 3n = 40

n2 - 3n - 40 = 0

(n + 5)(n - 8) = 0

so n = -5 or 8. As a polygon cannot have a negative number of sides, n = 8.

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Q: A polygon has 20 diagonals how many sides does it have?
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