Sum of Angles in a Polygon - Formula and Examples

What Is The Sum of Angles In A Polygon?

The sum of the interior angles of a polygon with $n$ sides is $(n-2) \times 180°$. That single formula covers every polygon - a triangle, a hexagon, a 100-sided shape - and it holds whether the polygon is regular (all sides and angles equal) or irregular (sides and angles of different sizes). An interior angle is the angle formed inside the polygon at each vertex, between two sides that meet there.

The number never changes for a given number of sides. Bend a quadrilateral into any four-sided shape you like, and its four interior angles still add to 360°. That fixedness is what makes the formula so useful.

Where Does The (n−2)·180° Formula Come From?

The formula is not something to memorise blind - it drops straight out of one move: cut the polygon into triangles.

Pick any single vertex of the polygon. Draw diagonals from that one vertex to every other non-adjacent vertex. This slices the polygon into a fan of triangles that don't overlap.

• A quadrilateral ($n = 4$) splits into 2 triangles. Sum $= 2 \times 180° = 360°$.

• A pentagon ($n = 5$) splits into 3 triangles. Sum $= 3 \times 180° = 540°$.

• A hexagon ($n = 6$) splits into 4 triangles. Sum $= 4 \times 180° = 720°$.

Count the triangles and a pattern appears: a polygon with $n$ sides always splits into exactly $n - 2$ triangles from a single vertex. Since each triangle's three angles sum to 180° (the triangle sum theorem), the whole polygon's interior angles sum to:

$$\text{Sum of interior angles} = (n - 2) \times 180°$$

The "$-2$" is simply the two sides that touch the vertex you drew from — those two sides don't produce their own triangle, so you always end up with two fewer triangles than sides.

What about one interior angle of a regular polygon?

For a regular polygon, every interior angle is equal, so divide the total by the number of sides:

$$\text{Each interior angle} = \frac{(n - 2) \times 180°}{n}$$

A regular hexagon: $\frac{720°}{6} = 120°$ per angle. This division step only works when the polygon is regular — for irregular polygons the total is fixed, but the individual angles can differ.

What Is The Sum Of the Exterior Angles?

Here is the surprise that trips up most students: the sum of the exterior angles of any polygon is always 360° - no matter how many sides it has. An exterior angle is the angle between one side and the extension of the next side.

Walk all the way around the boundary of the polygon and you turn through a full circle exactly once, so the turns add up to 360°. A triangle, a pentagon, a 20-sided shape - every one of them totals 360° in exterior angles. The interior sum grows with more sides; the exterior sum does not.

Examples of the Sum of Angles in a Polygon

Each example is worked step by step. The problem statement is in bold; the working is not.

Example 1

Find the sum of the interior angles of a hexagon.

A hexagon has $n = 6$ sides.

$$\text{Sum} = (6 - 2) \times 180°$$ $$= 4 \times 180°$$ $$= 720°$$

Final answer: 720°.

Example 2

A student is told a regular octagon's interior angles sum to 8 × 180° = 1440°. Is that right?

The tempting move is to multiply the number of sides straight by 180°, because "each side has an angle."

Wrong path: $8 \times 180° = 1440°$.

Check it against a shape you know. A quadrilateral has 4 sides. By the same wrong logic, $4 \times 180° = 720°$ - but a quadrilateral (a square, say) has four right angles summing to $4 \times 90° = 360°$, not 720°. So the shortcut fails.

The fix is the "$-2$." An octagon splits into $8 - 2 = 6$ triangles.

$$\text{Sum} = (8 - 2) \times 180° = 6 \times 180° = 1440°$$

The final number happens to look similar, but the reasoning is different: it is $6 \times 180°$, not $8 \times 180°$. Skipping the "$-2$" gives $1440°$ here only by coincidence of arithmetic - for a pentagon it would give $900°$ instead of the correct $540°$. Always subtract 2 first.

Final answer: 1440°, from $(8-2) \times 180°$.

Example 3

Find each interior angle of a regular pentagon.

First the total: $n = 5$, so sum $= (5 - 2) \times 180° = 3 \times 180° = 540°$.

A regular pentagon has 5 equal angles, so divide:

$$\text{Each angle} = \frac{540°}{5} = 108°$$

Final answer: 108° per interior angle.

Example 4

A polygon's interior angles sum to 1080°. How many sides does it have?

Set the formula equal to 1080° and solve for $n$:

$$(n - 2) \times 180° = 1080°$$ $$n - 2 = \frac{1080°}{180°} = 6$$ $$n = 8$$

Final answer: The polygon has 8 sides - it is an octagon.

Example 5

An irregular quadrilateral has three angles measuring ∠A = 80°, ∠B = 100°, and ∠C = 95°. Find ∠D.

The sum for any quadrilateral is $(4 - 2) \times 180° = 360°$, regular or not.

$$\angle A + \angle B + \angle C + \angle D = 360°$$ $$80° + 100° + 95° + \angle D = 360°$$ $$275° + \angle D = 360°$$ $$\angle D = 85°$$

Final answer: ∠D = 85°. The fixed total is what lets you find a missing angle.

Example 6

A regular polygon has each exterior angle equal to 24°. How many sides does it have, and what is its interior-angle sum?

Since exterior angles of any polygon sum to 360°, and this one is regular:

$$n = \frac{360°}{24°} = 15$$

Now the interior sum:

$$\text{Sum} = (15 - 2) \times 180° = 13 \times 180° = 2340°$$

Final answer: 15 sides, interior-angle sum 2340°.

Why This Formula Matters - "Every shape must close on itself"

The interior-angle sum is not a classroom curiosity. It is a closure condition: for a many-sided shape to close up into a loop with no gap and no overlap, its angles are forced to a fixed total.

This is why the formula shows up wherever shapes have to fit:

• Tiling and floors - square tiles (four 90° corners = 360°) meet perfectly around a point. Regular pentagons (108° each) cannot tile a flat floor, because 108° does not divide 360° evenly. The angle sum decides what can and cannot cover a surface.

• Structural design - the geodesic domes made famous by the engineer and architect Buckminster Fuller rely on knowing exactly how the angles of many small polygons add up so the whole surface curves and closes.

• Nature - a honeycomb uses hexagons (720° total, 120° each) because that shape tiles the plane while using the least wax for the most storage.

The deeper idea, worth carrying forward: the interior sum grows with the number of sides, but the exterior sum stays pinned at 360° forever - because turning around any closed loop is always one full rotation, whatever its shape. That single fact is the seed of a whole branch of mathematics called topology.

What Are The Most Common Mistakes With Polygon Angle Sums?

Mistake 1: Forgetting to subtract 2

Where it slips in: The moment a student reaches for the formula and multiplies the number of sides straight by 180°.

Don't do this: For a pentagon, writing $5 \times 180° = 900°$.

The correct way: Subtract 2 first - a pentagon splits into $5 - 2 = 3$ triangles, so $(5 - 2) \times 180° = 540°$. The first instinct on any polygon-sum question is to multiply the side count by 180° and skip the "$-2$"; anchoring the formula to the number of triangles (always two fewer than the sides) is what stops that.

Mistake 2: Confusing the interior sum with one interior angle

Where it slips in: Regular-polygon questions that ask for "each angle," not the total.

Don't do this: Answering "540°" when asked for one angle of a regular pentagon.

The correct way: The total is 540°; divide by the number of sides to get one angle: $540° \div 5 = 108°$. Read whether the question wants the sum or one angle - the memorizer who learned only "$(n-2) \times 180°$" without the division step gives the whole total when a single angle was asked for.

Mistake 3: Mixing up interior and exterior angle sums

Where it slips in: Any problem that gives an exterior angle and asks about the shape.

Don't do this: Using $(n-2) \times 180°$ when the question is about exterior angles.

The correct way: The exterior angles of every polygon sum to 360°, a value that does not depend on the number of sides. Keep the two facts separate: interior sum grows, exterior sum is fixed. This exact mistake has an expensive real-world cousin - surveyors closing a boundary traverse check that their measured exterior turning angles add to exactly 360°; a running total that drifts from 360° signals a measurement error before the plot is ever drawn, the same way a mis-added polygon fails to close on paper.

Conclusion

• The sum of angles in a polygon (interior) is $(n - 2) \times 180°$ for any polygon with $n$ sides, regular or irregular.

• The formula comes from splitting the polygon into $n - 2$ triangles, each contributing 180°.

• Each interior angle of a regular polygon is $\frac{(n-2) \times 180°}{n}$.

• The sum of exterior angles is always 360°, regardless of the number of sides.

• A fixed total lets you find a missing angle or work backward to the number of sides.

To take polygon angles further with a teacher, explore Bhanzu's geometry tutor or middle school math tutor programs, or browse math classes online.

A Practical Next Step

Practice these problems to solidify your understanding. Work out the interior-angle sum of a decagon (10 sides), then find each angle assuming it is regular. Next, given an interior-angle sum of 1620°, find the number of sides. If the "$-2$" trips you up, return to the section on where the formula comes from and re-draw the triangle split. Want a live Bhanzu trainer to walk through more polygon problems? Book a free demo class.

Read More

• Interior angles of a polygon

• Exterior angles of a polygon

• Types of polygon

• Types of angles in geometry

• What is a polygon

• Quadrilaterals and their angles