Question

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

Answer 1

`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^@`