Mathematics · Euclidean Geometry

Polygon Perimeter (Geometry)

Reference entry · last updated September 13, 2026

Polygon perimeter denotes the total linear distance around the boundary of a closed two-dimensional polygon, defined as the sum of all its consecutive side lengths [1].

1. First principles and general definition

A polygon in the Euclidean plane is bounded by a finite sequence of straight line segments called edges or sides. For an $n$-sided polygon with side lengths $s_1, s_2, \dots, s_n$, the perimeter $P$ is the scalar sum of all edge lengths [1]:

$$P = \sum_{i=1}^{n} s_i = s_1 + s_2 + \dots + s_n$$

Given the coordinates of vertices $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$ in cyclic order, the perimeter evaluates as:

$$P = \sum_{i=1}^{n} \sqrt{(x_{i+1} - x_i)^2 + (y_{i+1} - y_i)^2}, \quad \text{where } (x_{n+1}, y_{n+1}) = (x_1, y_1)$$

2. Regular polygons and equilateral figures

A polygon is regular when all $n$ sides are equal in length and all interior angles are congruent. Let $s$ denote the common side length [1]:

$$P = n \cdot s$$

If a regular $n$-gon has circumradius $R$ (distance from center to each vertex) or inradius / apothem $r$ (distance from center to edge midpoint):

$$s = 2R \sin\left(\frac{\pi}{n}\right) = 2r \tan\left(\frac{\pi}{n}\right)$$

$$P = 2nR \sin\left(\frac{\pi}{n}\right) = 2nr \tan\left(\frac{\pi}{n}\right)$$

As $n \to \infty$ with fixed circumradius $R$, $P \to 2\pi R$, recovering the circumference of a circle.

3. Standard quadrilaterals

For four-sided polygons ($n = 4$), geometric constraints simplify perimeter computation [1, 2]:

Square: Four congruent sides of length $s$:
$$P = 4s$$
Rectangle / Parallelogram: Adjacent sides of length $l$ and $w$:
$$P = 2(l + w) = 2l + 2w$$
Rhombus: Four congruent sides of length $s$:
$$P = 4s$$
Trapezoid: Two parallel bases $b_1, b_2$ and two non-parallel legs $c, d$:
$$P = b_1 + b_2 + c + d$$

4. Perimeter of composite shapes

A composite shape is formed by joining two or more standard geometric figures along shared boundary segments. The perimeter of a composite shape accounts strictly for the exterior boundary [1].

When two shapes $A$ and $B$ share an internal interface of length $L_{\text{shared}}$, the resulting composite perimeter satisfies:

$$P_{\text{composite}} = P_A + P_B - 2 \cdot L_{\text{shared}}$$

Internal dividing segments must not be added to the perimeter calculation because they do not form part of the outer perimeter boundary.

References

^H.S.M. Coxeter, Introduction to Geometry, 2nd ed., John Wiley & Sons, 1969.
^Euclid, Elements, Book I, Definition 22 (Classification of Quadrilaterals).