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740

The number of digaonals is given by n(n-3)/2, where n is the number of sides (or vertices) of a polygon:

40(40-3)/2=20*37=740

For a proof, see:

http://www.artofproblemsolving.com/Wiki/index.php/Diagonal

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Q: How many diagonals will a 40 sided polygon have?
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How many diagonals are there in a 40-sided polygon?

0.5*(402-120) = 740 diagonals


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Does a 40 sided polygon have 6840 diagonals?

The sum of the angles of a regular n-sided polygon is equal to (n - 2) x 180 degrees.Therefore, the sum of the angles of a regular 40-sided polygon (tetradecagon) is equal to (40 - 2) x 180 = 6840 degrees. Therefore, 6840 refers to the sum of the degrees, not the number of diagonals.The number of diagonals of an n-sided polygon is given by n(n-3)/2. So a 40-sided polygon has 40*37/2=740 diagonals.


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