Each interior angle of a regular polygon lies between 136 to 142. How do we calculate the sides of the polygon?

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Answer 1 · 2016-01-07T00:51:21

#n=9#

In a regular polygon each interior angle can be obtained in this way:
#alpha=180^@-360^@/n#

From the conditions of the problem:
#136^@ < alpha<142^@#

That's the conjugation of this two inequations:
#136^@ < alpha# and #alpha<142^@#

Resolving the first inequation
#136^@<180^@-360^@/n#
#136^@<(180^@*n-360^@)/n# => #136^@*n<180^@.n-360^@# => #44^@.n>360^@# => #n>8.18#

Resolving the second inequation
#180^@-360^@/n<142^@#
#180^@*n-360^@<142^@*n# => #38^@.n<360^@# => #n<9.47#

Conjugating the two inequations
#8.18 < n<9.47#

Since #n in NN#, its only value that satisfies the inequation is #n=9#

By the way
#alpha=180^@-360^@/9=180^@-40^@# => #alpha=140^@#