POP

G4.11 Angles in Polygons

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01 — Explanation

What's the difference between a regular and an irregular polygon?

A regular polygon has all sides and all angles equal, so one angle can be found by simple division. An irregular polygon doesn't — and crucially, you should never assume a polygon is regular unless the question states it or the diagram marks equal sides/angles explicitly.

02 — Explanation

How do I find the sum of the interior angles of any polygon?

Any $n$ -sided polygon can be split into $(n-2)$ triangles by drawing diagonals from one vertex, and each triangle contributes $180°$ :

\text{Sum of interior angles} = (n-2)\times180°

Exam Tip

The most common slip is forgetting the $-2$ and using $n\times180°$ instead. Check your formula against a shape you know by heart — a quadrilateral ( $n=4$ ) must give $360°$ : $(4-2)\times180=360$ ✓.

For an irregular polygon, find the total using this formula, then subtract the sum of all the known angles to isolate the missing one.

03 — Explanation

Why is the exterior angle approach often easier for regular polygons?

The exterior angles of any convex polygon always sum to exactly $360°$ , regardless of how many sides it has. For a regular polygon, each exterior angle is therefore:

\text{Exterior angle} = \frac{360°}{n}

Since interior and exterior angles at the same vertex always sum to $180°$ (they lie on a straight line), the interior angle follows immediately:

\text{Interior angle} = 180° - \text{Exterior angle}

This two-step route is usually faster and less error-prone than the direct interior-angle formula $\dfrac{(n-2)\times180°}{n}$ .

04 — Explanation

How do I find the number of sides from a given angle?

If you're given the exterior angle: $n = \dfrac{360°}{\text{exterior angle}}$ . If you're given the interior angle instead, convert it to the exterior angle first ( $180°$ minus it), then divide into $360°$ . A polygon with an exterior angle of $40°$ has $n=360\div40=9$ sides (a nonagon).

Exam Tip

A frequent error is mixing up which angle you've calculated — if your answer comes out obtuse (over $90°$ ), it's almost certainly the interior angle; if it's acute, it's the exterior angle. Always sanity-check against this before writing a final answer.

05 — Explanation

How do I combine polygon angles with other angle facts?

Multi-mark questions often place a regular polygon next to an irregular shape, linked by a straight line. Work through the diagram step by step: find the regular polygon's interior angle first, use it inside the adjacent irregular shape's angle sum, then finish with a straight-line or vertically-opposite fact to reach the final unknown angle — annotating each value directly onto the diagram as you go keeps the chain of reasoning visible and secures reasoning marks.

06 — Explanation

What's a quick way to name common polygons by their number of sides?

Sides ( $n$ )	Name	Sum of interior angles
3	Triangle	$180°$
4	Quadrilateral	$360°$
5	Pentagon	$540°$
6	Hexagon	$720°$
8	Octagon	$1080°$
10	Decagon	$1440°$

Recognising these by name helps you sanity-check an answer quickly — if a question describes a hexagon and your interior angle sum doesn't come out as $720°$ , you know to recheck your working before moving on.

07 — Worked example

Worked example

A regular polygon has an interior angle of $156°$ . Find the number of sides, $n$ , and name the polygon.

Find the exterior angle: $180-156=24°$ .
$n=360\div24=15$ .
The polygon is a regular 15-sided shape (a pentadecagon).

Check: interior angle via the direct formula $=\dfrac{(15-2)\times180}{15}=\dfrac{2340}{15}=156°$ ✓.