Exterior Angles of a Polygon: Sum & Formula

What Is an Exterior Angle of a Polygon?

An exterior angle of a polygon is the angle formed outside the polygon between one side and the extension of the side next to it. At every vertex, the side continues past the corner as a straight line, and the exterior angle is the angle between that extension and the following side.

Each vertex has an exterior angle that pairs with the interior angle sitting inside the polygon there. Because the side and its extension form a straight line, the two together make a straight angle:

$$\text{interior angle} + \text{exterior angle} = 180°.$$

So the exterior angle is the supplement of the interior angle at the same vertex — it is "how much the direction of travel turns" as you round that corner. For the inside-the-shape companion to this idea, see the interior angles article; for shapes in general, the polygon overview.

Why the Exterior Angles Always Sum to $360°$

This is the result everything else depends on, so it is worth deriving rather than memorising. Take any polygon with $n$ sides.

Each vertex contributes one interior and one exterior angle, and the two add to $180°$. With $n$ vertices, the interior and exterior angles together come to:

$$n \times 180°.$$

Now subtract the part that is interior. The sum of the interior angles of an $n$-sided polygon is $(n - 2) \times 180°$ — the result derived in the interior angles article by slicing the polygon into $(n-2)$ triangles. What is left over is the sum of the exterior angles:

$$\text{sum of exterior angles} = n \times 180° - (n - 2) \times 180°.$$

Expand the bracket: $(n - 2) \times 180° = 180°n - 360°$. So:

$$\text{sum of exterior angles} = 180°n - (180°n - 360°) = 360°.$$

The $180°n$ terms cancel, and $360°$ is all that remains — with no $n$ in it. That is why the sum does not depend on the number of sides, and why it holds for regular and irregular polygons alike. The walking-around-the-field picture is the same truth without the algebra: one trip around the boundary is one full turn.

The Exterior Angle of a Regular Polygon

A regular polygon has all sides equal and all angles equal, so its exterior angles are all the same size. Since they share a total of $360°$ equally among $n$ vertices, each exterior angle is:

$$\text{each exterior angle} = \frac{360°}{n}.$$

A reader question that comes up constantly: what is the exterior angle of a regular hexagon? With $n = 6$, each is $\dfrac{360°}{6} = 60°$. A square ($n = 4$) gives $90°$; an equilateral triangle ($n = 3$) gives $120°$.

This formula only works for regular polygons, because only there are the exterior angles equal. For an irregular polygon, the angles differ, but they still total $360°$ — so you find a missing exterior angle by subtracting the known ones from $360°$.

The formula also runs backwards. If you know each exterior angle of a regular polygon, the number of sides is $n = \dfrac{360°}{\text{each exterior angle}}$ — a quick way to identify a shape from one angle.

Examples of Exterior Angles of a Polygon

With the $360°$ sum and the regular-polygon formula in hand, here is the topic doing real work. The problems build from a direct division up to finding the number of sides.

Example 1

Find each exterior angle of a regular octagon.

A regular octagon has $n = 8$ sides: each exterior angle is $\dfrac{360°}{8} = 45°$.

Example 2

Find each interior angle of a regular pentagon using its exterior angle.

Wrong attempt. A student reasons: "the exterior angle is $\dfrac{360°}{5} = 72°$, and interior plus exterior is $360°$, so the interior angle is $360° - 72° = 288°$." That uses the wrong total.

Why it breaks. An interior angle of a closed pentagon of $288°$ is impossible — it is a reflex angle, and a convex polygon's corners are not reflex. The error is pairing interior and exterior against $360°$ instead of $180°$. The $360°$ is the total of all the exterior angles; the interior–exterior pair at one vertex sits on a straight line, so it sums to $180°$.

Correct. Each exterior angle is $\dfrac{360°}{5} = 72°$. The interior angle is its supplement: $180° - 72° = 108°$. (Check against the interior-angle formula: $\dfrac{(5-2)\times 180°}{5} = \dfrac{540°}{5} = 108°$ — it matches.)

Example 3

Three exterior angles of a quadrilateral are $80°$, $95°$, and $70°$. Find the fourth.

The four exterior angles sum to $360°$: the fourth is $360° - (80° + 95° + 70°) = 360° - 245° = 115°$.

Example 4

A regular polygon has each exterior angle equal to $40°$. How many sides does it have?

Use $n = \dfrac{360°}{\text{each exterior angle}} = \dfrac{360°}{40°} = 9$. It is a regular nonagon.

Example 5

A regular polygon has each interior angle $150°$. Find each exterior angle and the number of sides.

Each exterior angle is the supplement of the interior angle: $180° - 150° = 30°$. Then $n = \dfrac{360°}{30°} = 12$. It is a regular dodecagon.

Example 6

Can a regular polygon have an exterior angle of $50°$?

For a regular polygon, $n = \dfrac{360°}{50°} = 7.2$. The number of sides must be a whole number, and $7.2$ is not — so no regular polygon has an exterior angle of exactly $50°$. The exterior angle of a regular polygon must divide $360°$ evenly.

Why Exterior Angles Matter Beyond the Classroom

The constant $360°$ turn is not a textbook curiosity — it is the rule that anything following a closed path obeys.

• Robotics and turtle geometry. Programming a robot to trace a polygon means telling it to turn by the exterior angle at each corner. To draw a regular hexagon it turns $60°$ six times — $360°$ total — and arrives home facing the start.

• Surveying and land boundaries. A surveyor walking a property boundary records the turn at each corner; those turns must close to $360°$, and a total that misses it flags a measurement error before the plot is filed.

• Road and track design. A closed racetrack or a roundabout is a loop; the turning angles around it sum to one full circle, which sets how the banking and sightlines are planned.

• Tiling and structure. Knowing each exterior angle of a regular polygon is how designers check which shapes meet cleanly at a point — the same reasoning that tells you hexagons tile a floor and pentagons do not.

For a Grade 8 student, the exterior-angle sum is the first time a geometric total stays fixed while the shape changes wildly — the same $360°$ for a triangle and a thousand-gon — and that invariance is a small taste of how powerful a single rule can be.

Where Students Trip Up on Exterior Angles

Mistake 1: Pairing interior and exterior against $360°$ instead of $180°$

Where it slips in: Converting between an interior and an exterior angle at the same vertex, the student subtracts from $360°$.

Don't do this: Compute "interior $= 360° -$ exterior" at one vertex.

The correct way: At a single vertex the side and its extension form a straight line, so interior $+$ exterior $= 180°$. The $360°$ is the total of all the exterior angles across the whole polygon — two different totals.

Mistake 2: Using $\dfrac{360°}{n}$ on an irregular polygon

Where it slips in: A problem shows an irregular polygon and the student divides $360°$ by the number of sides to find "each" exterior angle.

Don't do this: Claim every exterior angle of an irregular pentagon is $72°$.

The correct way: Only regular polygons have equal exterior angles, so $\dfrac{360°}{n}$ applies only there. For an irregular polygon the angles differ; the sum is still $360°$, so find a missing one by subtracting the known angles from $360°$. The memorizer who learned "$360$ over $n$" without the word regular gets caught on every irregular figure.

Mistake 3: Treating the sum as depending on the number of sides

Where it slips in: A student expects a many-sided polygon to have a larger exterior-angle sum, by analogy with the interior-angle sum that does grow with $n$.

Don't do this: Write "sum of exterior angles $= (n-2) \times 180°$" or otherwise let $n$ into the sum.

The correct way: The exterior-angle sum is always $360°$ — the $n$ cancels in the derivation. It is the interior sum that grows with $n$; the exterior sum is fixed. Keep the two results separate.

Key Takeaways

• The exterior angles of a polygon always sum to $360°$, regardless of the number of sides or whether the polygon is regular.

• That sum comes from $n \times 180°$ minus the interior-angle sum $(n-2)\times 180°$ — the $n$ cancels, leaving $360°$.

• For a regular polygon, each exterior angle is $\dfrac{360°}{n}$, and the number of sides is $\dfrac{360°}{\text{each exterior angle}}$.

• At any vertex, interior $+$ exterior $= 180°$ — they are supplementary.

• The most common mistake is mixing the per-vertex $180°$ with the whole-polygon $360°$, or using $360°/n$ on an irregular polygon.

Practice These Problems to Solidify Your Understanding

1. Find each exterior angle of a regular decagon (10 sides).

2. A regular polygon has each exterior angle $24°$. How many sides does it have?

3. Four exterior angles of a pentagon are $60°$, $75°$, $80°$, and $65°$. Find the fifth.

Answer to Question 1: $36°$. Answer to Question 2: $15$ sides. Answer to Question 3: $360° - (60° + 75° + 80° + 65°) = 360° - 280° = 80°$. If Question 3 used a sum other than $360°$, revisit Mistake 1.

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