Q:

How many diagonals does a dodecagon have?

A:

A dodecagon, a polygon with 12 sides and 12 vertices, has 54 distinct diagonals. The formula for determining the number of diagonals of an n-sided polygon is n(n - 3)/2; thus, a dodecagon has 12(12 - 3)/2 = 12(9)/2 = 108/2 = 54 diagonals.

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Credit: Katie Walker CC-BY-2.0

The diagonal of a polygon is any line segment joining two nonadjacent vertices. A dodecagon has 12 vertices, and each vertex runs to the middle of the polygon to connect to nine other vertices. Thus, each vertex forms nine diagonals. There are three fewer diagonals than there are vertices. This is the first part of the procedure for finding the number of diagonals of a polygon: n(n - 3) = 12 x 9 = 108. Each diagonal has two end points, so in order to not count a duplicate diagonal, the final step is to divide 108 by 2, which results in 54 diagonals.

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Related Questions

• A:

A nonagon, or enneagon, is a polygon with nine sides and nine vertices, and it has 27 distinct diagonals. The formula for determining the number of diagonals of an n-sided polygon is n(n - 3)/2; thus, a nonagon has 9(9 - 3)/2 = 9(6)/2 = 54/2 = 27 diagonals.

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An octagon has 20 diagonals. A shape's diagonals are determined by counting its number of sides, subtracting three and multiplying that number by the original number of sides. This number is then divided by two to equal the number of diagonals.

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A hexagon has nine diagonals regardless of whether it is a regular or irregular polygon. This can be determined by utilizing a simple formula that applies to polygons with any number of sides.