What's the exterior angle of a regular polygon?

The exterior angles of any polygon always add up to 360°, so for a regular polygon with n sides each exterior angle is 360 ÷ n. A regular pentagon has 5 sides, so each exterior angle is 360 ÷ 5 = 72°. A regular hexagon has 6 sides, so each is 360 ÷ 6 = 60°. The 360° total comes from walking once around the shape: at each corner you turn through the exterior angle, and one full lap is a 360° turn.

How do I find the interior angle once I have the exterior angle?

An interior angle and its exterior angle sit on a straight line, so they add to 180°. Find the exterior angle first, then subtract from 180°. For a regular pentagon the exterior angle is 360 ÷ 5 = 72°, so each interior angle is 180 − 72 = 108°. For a regular hexagon it is 180 − 60 = 120°. Going straight for the interior angle by dividing 180 by the number of sides does not work — that ignores the 360° rule for the exterior angles.

Why does a hexagon have a smaller exterior angle than a pentagon?

Because the same 360° is shared between more corners. A pentagon shares 360° over 5 corners (72° each); a hexagon shares the same 360° over 6 corners (60° each). More sides means more corners to split the fixed 360° between, so each exterior angle is smaller, not bigger. It is the interior angle that grows with more sides — the exterior angle shrinks.

GCSE Maths Foundation · AQA · Angles, shape & trigonometry

Angles in polygons: exterior angles sum to 360°, not 180°

Polygon angles trip students who borrow the triangle fact — that the angles of a triangle add to 180° — and assume the exterior angles of any polygon must add to 180° too. They do not. The exterior angles of every polygon always sum to 360°, so each exterior angle of a regular $n$ -gon is $360 \div n$ .

The thirty-second fix: the exterior angles of any polygon sum to 360°, so each exterior angle of a regular $n$ -gon is $360 \div n$ , and the interior angle is $180^\circ$ minus that. A regular pentagon: exterior $360 \div 5 = 72^\circ$ , interior $180 - 72 = 108^\circ$ . A regular hexagon: exterior $360 \div 6 = 60^\circ$ — smaller, not larger.

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How to spot it in your own work

You divided $180$ by the number of sides (e.g. $180 \div 5 = 36^\circ$ ), as if the exterior angles summed to 180° like a triangle’s interior angles.
You found the exterior angle but forgot to subtract from 180° to get the interior angle, reporting $72^\circ$ where the question wanted $108^\circ$ .
You assumed more sides means a bigger exterior angle, when more sides splits the fixed 360° further, making each one smaller.
You mixed up which sum is which — interior angles do not add to a fixed number across polygons, but the exterior angles always total 360°.

An exam question that triggers it

Here is a canonical AQA Foundation trigger (regular polygon angle, either paper):

A regular pentagon has 5 equal sides and 5 equal angles.

Work out the size of one interior angle.

The misconception is to treat the exterior angles like a triangle’s interior angles and divide $180 \div 5 = 36^\circ$ , or to find the exterior angle and stop there at 72° without subtracting.

The exterior angles of any polygon sum to 360°, so each exterior angle is $360 \div 5 = 72^\circ$ , and the interior angle is the straight-line partner: $180 - 72 = 108^\circ$ .

Why students fall for this

The very first angle fact a student locks in is that a triangle’s angles add to 180°. It is so well-drilled that the number 180 becomes a reflex for “angles in a shape”, and it gets carried onto polygons in general. So when a pentagon’s angles come up, the student divides 180 between the corners and lands on 36° — the triangle’s number applied to the wrong shape and the wrong kind of angle.

The meaning fixes it. The fixed total is for the exterior angles, and it is 360°, because walking once around the outside of any polygon turns you through a full circle: at each corner you turn by the exterior angle, and after one lap you have turned 360° in total. Share that 360° equally over $n$ corners and each exterior angle is $360 \div n$ . The interior angle sits on a straight line with its exterior angle, so it is $180^\circ$ minus the exterior angle.

AQA Foundation papers exploit this directly: find an interior angle of a regular polygon (exterior $360 \div n$ , then $180 -$ that), or work back from an exterior angle to the number of sides ( $n = 360 \div$ exterior). The comparison between a pentagon and a hexagon is the giveaway that disproves “more sides, bigger angle”.

Worked example — pentagon versus hexagon. Compare one exterior angle of a regular pentagon with one exterior angle of a regular hexagon. Which is bigger?

Share the fixed 360° over each shape’s corners:

\text{pentagon: } 360 \div 5 = 72^\circ \qquad \text{hexagon: } 360 \div 6 = 60^\circ

The hexagon’s exterior angle is smaller ( $60^\circ < 72^\circ$ ), because the same 360° is split between 6 corners instead of 5. More sides means a smaller exterior angle — it is the interior angle ( $180 - 60 = 120^\circ$ versus the pentagon’s $108^\circ$ ) that grows.

The fix: Exterior angles sum to 360°: each is 360 ÷ n, and the interior angle is 180° minus it

Start from the 360° fact. The exterior angles of any polygon always add to 360°, whatever the number of sides. This is the one fixed total, not 180°.

Divide for one exterior angle. For a regular $n$ -gon, each exterior angle is $360 \div n$ . Pentagon: $360 \div 5 = 72^\circ$ . Hexagon: $360 \div 6 = 60^\circ$ .

Subtract from 180° for the interior angle. The interior and exterior angles sit on a straight line, so they add to 180°. Pentagon: $180 - 72 = 108^\circ$ . Hexagon: $180 - 60 = 120^\circ$ .

Expect the exterior angle to shrink with more sides. The fixed 360° is shared over more corners, so each exterior angle gets smaller as $n$ grows. If a polygon with more sides gave you a bigger exterior angle, you have the rule backwards.

Worked example

A regular polygon has an exterior angle of 40°. How many sides does it have, and what is each interior angle? This is the work-backwards form: the 360° rule still does the lifting.

Use the 360° total. The exterior angles sum to 360°, so the number of sides is
$n = 360 \div 40 = 9 \text{ sides}$
Find the interior angle. It is the straight-line partner of the exterior angle:
$180 - 40 = 140^\circ$
Check it against the trap. Dividing $180 \div 9 = 20^\circ$ would treat the exterior angles as summing to 180° — wrong total, wrong answer. The right total is 360°.
Sanity-check the direction. Nine sides is more than a hexagon’s six, and its exterior angle (40°) is indeed smaller than the hexagon’s 60° — more sides, smaller exterior angle, as it should be.

The same 360° rule settles the trigger: a regular pentagon has exterior angle $360 \div 5 = 72^\circ$ , so each interior angle is $180 - 72 = 108^\circ$ , never the $180 \div 5 = 36^\circ$ the triangle reflex suggests.

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Common questions

What's the exterior angle of a regular polygon?: The exterior angles of any polygon always add up to 360°, so for a regular polygon with $n$ sides each exterior angle is $360 \div n$ . A regular pentagon has 5 sides, so each exterior angle is $360 \div 5 = 72^\circ$ . A regular hexagon has 6 sides, so each is $360 \div 6 = 60^\circ$ . The 360° total comes from walking once around the shape: at each corner you turn through the exterior angle, and one full lap is a 360° turn.
How do I find the interior angle once I have the exterior angle?: An interior angle and its exterior angle sit on a straight line, so they add to 180°. Find the exterior angle first, then subtract from 180°. For a regular pentagon the exterior angle is $360 \div 5 = 72^\circ$ , so each interior angle is $180 - 72 = 108^\circ$ . For a regular hexagon it is $180 - 60 = 120^\circ$ . Going straight for the interior angle by dividing 180 by the number of sides does not work — that ignores the 360° rule for the exterior angles.
Why does a hexagon have a smaller exterior angle than a pentagon?: Because the same 360° is shared between more corners. A pentagon shares 360° over 5 corners ( $72^\circ$ each); a hexagon shares the same 360° over 6 corners ( $60^\circ$ each). More sides means more corners to split the fixed 360° between, so each exterior angle is smaller, not bigger. It is the interior angle that grows with more sides — the exterior angle shrinks.

Related misconceptions

Trigonometry & Pythagoras: ratio choice and shorter sidesThe neighbouring right-angle skill: label opp/adj/hyp before picking sin/cos/tan, and subtract the squares for a side shorter than the hypotenuse.
Symmetry & 2D/3D shape: lines of symmetry and projectionsReading a shape by its real properties, not its appearance: counting lines of symmetry by folding and drawing plan and elevation as flat 2D views.

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