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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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

$n = 9$ $n=9$

Explanation:

In a regular polygon each interior angle can be obtained in this way:
$α = 180^{\circ} - \frac{360^{\circ}}{n}$ $alpha=180^@-360^@/n$

From the conditions of the problem:
$136^{\circ} < α < 142^{\circ}$ $136^@ < alpha<142^@$

That's the conjugation of this two inequations:
$136^{\circ} < α$ $136^@ < alpha$ and $α < 142^{\circ}$ $alpha<142^@$

Resolving the first inequation
$136^{\circ} < 180^{\circ} - \frac{360^{\circ}}{n}$ $136^@<180^@-360^@/n$
$136^{\circ} < \frac{180^{\circ} \cdot n - 360^{\circ}}{n}$ $136^@<(180^@*n-360^@)/n$ => $136^{\circ} \cdot n < 180^{\circ} . n - 360^{\circ}$ $136^@*n<180^@.n-360^@$ => $44^{\circ} . n > 360^{\circ}$ $44^@.n>360^@$ => $n > 8.18$ $n>8.18$

Resolving the second inequation
$180^{\circ} - \frac{360^{\circ}}{n} < 142^{\circ}$ $180^@-360^@/n<142^@$
$180^{\circ} \cdot n - 360^{\circ} < 142^{\circ} \cdot n$ $180^@*n-360^@<142^@*n$ => $38^{\circ} . n < 360^{\circ}$ $38^@.n<360^@$ => $n < 9.47$ $n<9.47$

Conjugating the two inequations
$8.18 < n < 9.47$ $8.18 < n<9.47$

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

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

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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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