Mathematics · Euclidean Geometry

Properties and Angles of Polygons

Reference entry · last updated September 20, 2026

A polygon is a closed plane figure bounded by a finite sequence of straight line segments that meet only at their endpoints. Its interior angle sum depends only on the number of sides, not on the shape: any simple \(n\)-gon has interior angles totaling \((n-2) \cdot 180^\circ\), and any convex \(n\)-gon has exterior angles totaling \(360^\circ\).

1. First principles and definitions

A polygon is determined by an ordered list of vertices \(V_1, V_2, \dots, V_n\) joined in sequence by the sides \(V_1V_2, V_2V_3, \dots, V_nV_1\). Three conditions make the figure a simple polygon: \(n \ge 3\), no two non-adjacent sides cross, and the boundary closes without gaps [1].

Convex polygon: every interior angle is less than \(180^\circ\), so every segment joining two interior points stays inside the polygon.
Concave polygon: at least one interior angle exceeds \(180^\circ\); the polygon has a reflex vertex.
Regular polygon: all sides are congruent and all interior angles are congruent.
Diagonal: a segment joining two non-adjacent vertices.

Every simple polygon can be triangulated into exactly \(n - 2\) triangles by drawing \(n - 3\) non-crossing diagonals. That count is the source of the interior angle formula.

2. Interior angle sum

For a convex \(n\)-gon, triangulating from one vertex produces \(n - 2\) triangles. Each triangle has interior angles summing to \(180^\circ\), and the triangles partition the polygon's interior angles, so the sums add [1, 2]:

\( S_{\text{interior}} = (n - 2) \cdot 180^\circ \)

The result is independent of convexity. Every simple polygon admits some noncrossing triangulation into \(n - 2\) triangles, so the same angle argument applies. For a concave polygon the triangulation may not be a fan from a single vertex, because such diagonals can leave the interior; a reflex vertex contributes an angle larger than \(180^\circ\), and the total still works out to \((n - 2) \cdot 180^\circ\).

Polygon	Sides \(n\)	Interior angle sum	Regular interior angle
Triangle	3	180°	60°
Quadrilateral	4	360°	90°
Pentagon	5	540°	108°
Hexagon	6	720°	120°
Octagon	8	1,080°	135°
\(n\)-gon	\(n\)	\((n-2) \cdot 180^\circ\)	\(\dfrac{(n-2) \cdot 180^\circ}{n}\)

3. Exterior angle sum

An exterior angle is formed by extending one side at a vertex and measuring the turn from that extension to the next side. At each vertex the interior and exterior angles are supplementary, so \( \text{interior} + \text{exterior} = 180^\circ \).

Summing over all \(n\) vertices and using the interior sum:

\( S_{\text{exterior}} = n \cdot 180^\circ - (n - 2) \cdot 180^\circ = 360^\circ \)

The exterior angle sum is \(360^\circ\) for every convex polygon regardless of the number of sides. It is also the total turning angle a traveler accumulates walking once around the boundary [2]. For a concave polygon, the signed exterior sum is still \(360^\circ\) when reflex turns are counted as negative.

4. Regular polygons

In a regular polygon every interior angle is equal, so each is the interior sum divided by \(n\), and each exterior angle is \(360^\circ / n\) [1]:

\( \text{interior angle} = \frac{(n-2) \cdot 180^\circ}{n}, \qquad \text{exterior angle} = \frac{360^\circ}{n} \)

With side length \(s\), the perimeter is \(P = ns\). The apothem \(a\), the perpendicular distance from center to a side, is \(a = \dfrac{s}{2 \tan(\pi / n)}\), which gives the regular polygon area:

\( A = \frac{1}{2} P a = \frac{n s^2}{4 \tan(\pi / n)} \)

As \(n\) grows without bound, the regular \(n\)-gon approaches a circle of the same perimeter, and its area approaches the circle's area.

5. Diagonals

Each of the \(n\) vertices can join \(n - 3\) non-adjacent vertices, and each diagonal is counted twice, so the number of distinct diagonals is:

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

A quadrilateral has \(2\) diagonals, a pentagon \(5\), a hexagon \(9\), and an octagon \(20\). The formula also confirms that a triangle has none.

6. Constraints and edge cases

A polygon requires \(n \ge 3\). Two or fewer segments cannot enclose an area.
Interior angles of a simple polygon are strictly between \(0^\circ\) and \(360^\circ\). A vertex with a \(180^\circ\) angle is collinear and produces a degenerate polygon.
For a convex polygon every interior angle is less than \(180^\circ\); a single angle of \(180^\circ\) or more makes it concave or degenerate.
The formulas use degrees by convention. In radians the interior sum is \((n-2)\pi\) and the exterior sum is \(2\pi\).
Congruent sides are not required for the angle-sum formulas. Only regularity is required for the equal-angle and equal-apothem results.

References

^Euclid, Elements, Books I and IV (Polygon definitions; regular polygon construction and angle results).
^H.S.M. Coxeter, Introduction to Geometry, 2nd ed., John Wiley & Sons, 1969.