The sum of all interior angles of a regular polygon is 1800°. How many diagonals does the polygon have?
54
The sum of the interior angles of an \(n\)-sided polygon is \((n-2)\times 180^\circ\).
Set this equal to 1800°: \((n-2)\times 180 = 1800\).
Solve: \(n - 2 = \dfrac{1800}{180} = 10 \Rightarrow n = 12\).
Number of diagonals \(= \dfrac{n(n-3)}{2} = \dfrac{12\times 9}{2} = 54\).
Hence, the polygon has 54 diagonals.
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