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How many sides are in a regular polygon that has exterior angles of 40°?

1 Answer
Jan 22, 2016

Answer:

A regular polygon with exterior angles of #40^o# would have 9 side and be a nonagon.

Explanation:

The exterior angles of any regular polygon must add up to #360^o#.

Since the angle measure given iin the questions s #40^o#, take #360^o/40^o# = 9. Meaning there are 9 exterior angles and therefore 9 sides to the polygon.

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A regular polygon refers to a multi-sided convex figure where all sides are equal in length and all angles have equal degree measures.
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The regular triangle has 3 interior angles of #60^o# and 3 exterior angles of #120^o#. The exterior angle have a sum of #360^o# #=(3)120^o#

The square has 4 interior angles of #90^o# and 4 exterior angles of #90^o#. The exterior angle have a sum of #360^o# #=(4)90^o#.

The square has 5 interior angles of #108^o# and 5 exterior angles of #72^o#. The exterior angle have a sum of #360^o# #=(5)72^o#.

In order to find the value of the interior angle of a regular polygon the equation is #((n-2)180)/n# where n is the number of sides of the regular polygon.

Triangle #((3-2)180)/3 = 60^o#
Square #((4-2)180)/4 = 90^o#
Pentagon #((5-2)180)/5 = 72^o#

Finally

The interior and exterior angles of a regular polygon form a linear pair and therefore are supplementary and must add up to #180^o#.