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

1 Answer
Jan 7, 2016

n=9

Explanation:

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^@