How many sides in this polygon?

**Motherof8** · January 16th 2010, 05:45 PM

The problem states: Find the number of sides in a regular polygon if the measure of an interior angle exceeds 6 times the measure of an exterior angle by 12. I wrote this formula: 180 - x = 6x + 12. In the back of the book, they say the answer is 15, but I get a different answer when I work it out. Do I have the wrong formula?

**Motherof8** · January 16th 2010, 05:51 PM

Never mind, I worked it out and found out the answer is 15.

**Soroban** · January 16th 2010, 06:21 PM

Hello, Motherof8!

According to your equation, $x$ = exterior angle.
Did you mean that?

Find the number of sides in a regular polygon if the measure of an interior angle
exceeds 6 times the measure of an exterior angle by 12.

Let: $x$ = interior angle.

Then: $180-x$ = exterior angle.

. . $\underbrace{\text{Interior }}_{x}\underbrace{\text{ is }}_{=} \underbrace{\text{ 6 times }}_{6\; \times}\underbrace{\text{ exterior }}_{(180-x)} \underbrace{\text{ plus 12}}_{ + \;12}$

We have: . $x \;=\;6(180-x) + 12 \quad\Rightarrow\quad x \;=\;1080 - 6x + 12 \quad\Rightarrow\quad 7x \:=\:1092$

Hence: . $x \:=\:156\quad\hdots$ each interior angle is 156°.

A regular polygon of $n$ sides has an interior angle of: / $180\,\frac{n-2}{n}$ degrees.

So we have: . $180\,\frac{n-2}{n} \:=\:156 \quad\Rightarrow\quad 180n - 360 \:=\:156n \quad\Rightarrow\quad 24n \:=\:260$

Therefore: . $n \,=\,15\quad\hdots$ The polygon has 15 sides.

Edit: You got it . . . Good for you!

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