Polygons and Interior Angles

A polygon is a closed plane figure whose boundary consists of finitely many straight segments. The usual angle-sum formulas apply to simple polygons, whose sides do not cross except at consecutive endpoints. A self-crossing star requires a different interpretation of interior and turning angles.

A polygon need not have equal sides or angles. Equal divisions of an angle sum are justified only when equal angles are given or the polygon is known to be regular.

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Convex and concave polygons

A convex polygon has every interior angle less than $180^\circ$ ; every segment joining two points in its region remains inside the region. A concave polygon has at least one reflex interior angle, greater than $180^\circ$ .

The boundary vertices are listed in order. In pentagon $ABCDE$ , for example, $AB$ and $BC$ are sides, while $AC$ joins non-adjacent vertices and is a diagonal. We exclude redundant straight-through vertices when counting the sides of the ordinary polygons discussed here.

A hexagon can be divided into four triangles

The triangle count is n − 2, not n. For n = 6, the total angle sum is 4 × 180° = 720°.

Diagram explanation and exact data

The diagonals from A to C, D and E split the hexagon into ABC, ACD, ADE and AEF. Internal diagonal angles partition the polygon angles, so adding the four triangle sums gives the polygon sum.

A hexagon can be divided into four triangles: exact reference data
Triangle	Vertices
1	A, B, C
2	A, C, D
3	A, D, E
4	A, E, F

Deriving the interior-angle sum

Draw diagonals from one vertex of a convex $n$ -gon to its non-adjacent vertices. They divide it into $n-2$ non-overlapping triangles. Each contributes $180^\circ$ , so

S=(n-2)180^\circ.

A hexagon divides into four triangles and has total angle measure $720^\circ$ . A decagon divides into eight triangles and has total $1440^\circ$ .

A simple concave polygon can also be triangulated, although drawing every diagonal from an arbitrary vertex may send some outside the polygon. A valid interior triangulation still has $n-2$ triangles, so the same sum holds. The proof needs non-overlapping triangles that cover the polygon, not merely any collection of diagonals.

Finding a missing angle

Four angles of a pentagon are $90^\circ$ , $80^\circ$ , $70^\circ$ and $70^\circ$ . The total is $540^\circ$ , leaving

540^\circ-(90^\circ+80^\circ+70^\circ+70^\circ)=230^\circ.

The reflex angle shows that this pentagon is concave. Replacing it with the smaller angle between the two sides would measure the wrong region.

Exterior angles are turns

For a convex polygon traversed once around its boundary, extend each side forward. The turn from one side’s direction to the next is the exterior angle. An interior angle $I$ and the corresponding turn $E$ satisfy

I+E=180^\circ.

After a complete circuit, the direction has turned through one full revolution, so the exterior turns total $360^\circ$ . This explains why the exterior sum does not increase with the number of sides.

For a concave polygon, some turns go the opposite way. A counterclockwise traversal has signed turns that sum to $360^\circ$ : a reflex interior angle of $230^\circ$ contributes a signed turn of $-50^\circ$ . Adding the positive magnitudes of all turns instead can give a different number. Do not transfer the convex unsigned rule without checking the convention.

When all angles are equal

An equiangular polygon has equal interior angles. A regular polygon has both equal angles and equal sides. Either condition of equal angles allows the interior sum to be divided by $n$ :

I=\frac{(n-2)180^\circ}{n},\qquad E=\frac{360^\circ}{n}.

For a regular nonagon, $n=9$ , so $I=140^\circ$ and $E=40^\circ$ . The formulas describe each angle, not the sum.

Recovering the number of sides

Suppose a convex regular polygon has interior angle $150^\circ$ . Its exterior turn is $30^\circ$ , so $n=360/30=12$ . It is a dodecagon.

If a proposed regular polygon has exterior angle $50^\circ$ , the calculation gives $n=7.2$ . A polygon cannot have a fractional number of sides, so the proposed exact angle does not define a regular polygon. The algebraic result is a consistency test, not a request to round $n$ .

Counting diagonals

From a vertex of an $n$ -gon, a diagonal can reach every vertex except itself and its two neighbours. That gives $n-3$ diagonals from each vertex. Multiplying by $n$ counts each diagonal from both ends, so

D=\frac{n(n-3)}2.

A hexagon has $6(3)/2=9$ diagonals. A decagon has $10(7)/2=35$ . These count all connections between non-adjacent vertices. In a concave polygon, some such diagonal segments lie outside the region; the combinatorial count is still the same.

The $n-3$ diagonals used in a triangulation are not all the polygon’s diagonals. They are a selected non-crossing set that divides the region into triangles.

Working from an angle sum

If a polygon’s interior-angle sum is $1260^\circ$ , solve

(n-2)180=1260,\qquad n-2=7,\qquad n=9.

This determines the number of sides but does not make the polygon regular. Its nine angles may differ while preserving the total.

For an irregular hexagon with five known angles, subtract their sum from $720^\circ$ . Do not use $720/6$ unless all six angles are known to be equal. The same distinction appears throughout geometry: a total relationship usually does not determine how that total is distributed.

Practice questions

What is the interior-angle sum of a heptagon?
A regular polygon has exterior angle $24^\circ$ . How many sides does it have?
How many diagonals does an octagon have?

Check your answers

A heptagon has seven sides, so the sum is $(7-2)180^\circ=900^\circ$ .
One full turn gives $n=360/24=15$ sides. Each interior angle is $156^\circ$ .
$D=8(8-3)/2=20$ . Dividing by two corrects the count from both endpoints.

Sources and further reading

Euclid, Elements I.32: the triangle angle-sum result used in triangulation.
AMSI, Parallelograms and Rectangles: polygon angle reasoning in quadrilaterals.