Angles and Polygons: Question 2

Worked solution

Key facts used

For any polygon with $n$ sides:

• The exterior angles always add up to $360^\circ$, so for a regular polygon each exterior angle is $\text{exterior angle} = \frac{360^\circ}{n}.$
• An interior angle and its adjacent exterior angle lie on a straight line, so $\text{interior} + \text{exterior} = 180^\circ.$
• The interior angles sum to $(n-2)\times 180^\circ$, so each interior angle of a regular polygon is $\text{interior angle} = \frac{(n-2)\times 180^\circ}{n}.$

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Part (a): find the number of sides from an interior angle

The interior angle is $150^\circ$, so the exterior angle is $180^\circ - 150^\circ = 30^\circ.$

The exterior angles of a regular polygon sum to $360^\circ$, so $n = \frac{360^\circ}{30^\circ} = 12.$

$\boxed{12 \text{ sides}}$

Check: a regular $12$-gon has interior angle $\dfrac{(12-2)\times180}{12} = \dfrac{1800}{12} = 150^\circ.$ ✓

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Part (b): find an interior angle from the number of sides

Method 1 (angle sum). For $n = 9$: $\text{sum of interior angles} = (9-2)\times 180^\circ = 7\times 180^\circ = 1260^\circ.$ Because the polygon is regular, all $9$ interior angles are equal: $\text{each interior angle} = \frac{1260^\circ}{9} = 140^\circ.$

Method 2 (via the exterior angle). Each exterior angle is $\dfrac{360^\circ}{9} = 40^\circ$, so each interior angle is $180^\circ - 40^\circ = 140^\circ.$

$\boxed{140^\circ}$

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Part (c): set up an equation linking interior and exterior angles

Let each exterior angle be $x^\circ$. Then each interior angle is $8x^\circ$, and they sit on a straight line: $8x + x = 180.$

Solve: $9x = 180 \quad\Rightarrow\quad x = 20.$

So each exterior angle is $20^\circ$. Using $n = \dfrac{360^\circ}{\text{exterior angle}}$: $n = \frac{360^\circ}{20^\circ} = 18.$

$\boxed{18 \text{ sides}}$

Check: interior angle $= 8\times 20^\circ = 160^\circ$, and $\dfrac{(18-2)\times180}{18} = \dfrac{2880}{18} = 160^\circ.$ ✓