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Equilateral polygon

An equilateral polygon is a polygon in which all sides are of equal length.^[1] Unlike a regular polygon, which requires both equal side lengths and equal interior angles, an equilateral polygon may have unequal angles.^[2] For instance, an equilateral triangle has all sides equal and is necessarily equiangular with each angle measuring 60 degrees, making it regular.^[3] In the case of quadrilaterals, a rhombus is an equilateral polygon where all four sides are congruent, but the angles are not necessarily equal unless it is a square.^[4] Equilateral polygons can be constructed for any number of sides n ≥ 3, and while regular examples are the most symmetric, non-regular equilateral n-gons exist for n ≥ 4, with increasing degrees of freedom in angle configurations as n grows.^[5] These polygons appear in various mathematical contexts, including lattice geometry, where their equal side lengths impose specific constraints on positioning and symmetry,^[6] and tiling problems.^[7]

Definition and Classification

Definition

A polygon is a plane figure formed by connecting a finite number of straight line segments end to end, with the endpoints of consecutive segments meeting at vertices to create a closed chain.^[8] An equilateral polygon is defined as a polygon in which all sides have the same length, typically denoted as a, independent of the interior angles at the vertices.^[1] This equality of side lengths distinguishes equilateral polygons from more general polygons, where side lengths may vary. The concept applies to simple polygons, encompassing both convex cases (where all interior angles are less than 180 degrees and the line segment between any two points inside lies entirely within the polygon) and concave cases (where at least one interior angle exceeds 180 degrees). For polygons with four or more sides, the definition extends to self-intersecting polygons, where sides cross each other, as seen in certain equilateral star configurations. Regular polygons represent a subset of equilateral polygons, characterized by both equal side lengths and equal interior angles.^[2]

Types of Equilateral Polygons

Equilateral polygons are classified primarily by their geometric configuration, including whether they are convex, concave, or self-intersecting, with specific constraints depending on the number of sides. Convex equilateral polygons have all interior angles less than 180° and no side intersections other than at vertices. For triangles (n=3), any equilateral polygon is necessarily convex and regular, as equal sides imply equal angles of 60° each. For n≥4, convex equilateral polygons exist that are not regular, allowing interior angles to vary while maintaining equal side lengths and satisfying the polygon closure condition; a representative example is the rhombus, where opposite angles are equal but adjacent angles differ unless it is a square. Concave equilateral polygons are simple polygons (non-self-intersecting) with at least one interior angle greater than 180°. Such configurations are impossible for n=3 or n=4, as equilateral triangles are always convex and equilateral quadrilaterals (rhombi) are always convex parallelograms. However, concave equilateral polygons exist for n≥5; for instance, the sphinx pentagon is a concave equilateral pentagon formed by assembling six equilateral triangles, resulting in one reflex angle.^[9] Self-intersecting equilateral polygons feature sides that cross each other, forming compound or star-like shapes while preserving equal side lengths. These include regular star polygons, such as the pentagram (Schläfli symbol {5/2}), where all sides are equal and the figure intersects itself in a symmetric manner. In general, for n≥4, the interior angles of an equilateral polygon can vary (subject to summing to (n-2)×180° and closure constraints) without requiring equality, distinguishing them from regular polygons; equilateral polygons are a related but distinct class from tangential polygons, which admit an incircle tangent to all sides.

Examples

Equilateral Triangles

An equilateral triangle is a polygon with three equal sides, and it is the simplest and most symmetric case of an equilateral polygon. Due to the properties of triangles, any equilateral triangle is necessarily equiangular, with all interior angles measuring exactly 60 degrees; this follows from the fact that equal sides imply equal base angles in an isosceles triangle, and since all three sides are equal, all angles are congruent, summing to 180 degrees as established by the angle sum theorem for triangles.^[10]^[11] This makes every equilateral triangle a regular polygon, a property unique to the triangular case among equilateral polygons, as higher-sided equilateral polygons can have unequal angles. The height h (or altitude) of an equilateral triangle with side length a is derived by drawing an altitude from one vertex to the opposite side, which bisects the base into two segments of length a/2 and forms two congruent 30-60-90 right triangles. Applying the Pythagorean theorem to one of these right triangles yields h = \sqrt{a^2 - (a/2)^2} = \frac{\sqrt{3}}{2} a.^[12] The area A is then computed as half the base times the height: A = \frac{1}{2} a \cdot h = \frac{\sqrt{3}}{4} a^2.^[12] Equilateral triangles exhibit high symmetry, with a rotational symmetry of order 3—allowing rotations by 120° and 240° about the centroid that map the triangle onto itself—and three lines of reflection symmetry along the altitudes. The complete symmetry group is the dihedral group D_3, consisting of these three rotations and three reflections, totaling six elements.^[13] This symmetry underscores the equilateral triangle's role as the foundational regular polygon. In the context of Viviani's theorem, the sum of the perpendicular distances from any interior point to the three sides equals the altitude h.^[14]

Equilateral Quadrilaterals

An equilateral quadrilateral, known as a rhombus, is defined as a quadrilateral with all four sides of equal length.^[15] In a rhombus, opposite angles are equal in measure, and consecutive angles are supplementary, summing to 180 degrees.^[15] These angle properties distinguish the rhombus from other parallelograms while allowing for variable configurations beyond the fixed angles of an equilateral triangle.^[16] The diagonals of a rhombus possess notable properties: they are perpendicular to each other, bisect each other at right angles, and each diagonal bisects the two opposite angles of the rhombus.^[17] These diagonal characteristics arise from the equal side lengths and contribute to the rhombus's symmetry.^[18] Rhombi exhibit variations in shape due to their flexible angles. A rhombus becomes a square, its regular form, when all interior angles measure 90 degrees.^[17] It can also flatten toward a degenerate parallelogram limit as one pair of angles approaches 0 degrees and the opposite pair approaches 180 degrees, reducing the area to zero.^[19] The area A of a rhombus with side length a and one interior angle \theta is given by the formula
A = a^2 \sin \theta.
This trigonometric expression leverages the height derived from the sine of the angle between adjacent sides.

Equilateral Polygons with Five or More Sides

Equilateral pentagons consist of five sides of equal length but can exhibit a range of interior angles, allowing for convex configurations that deviate from the regular form where all angles measure 108 degrees. These non-regular convex equilateral pentagons exist as a continuous family of shapes, classified by their central configurations in the plane, provided the angles sum to 540 degrees and the polygon closes without self-intersection. Unlike the regular pentagon, which is constructible with straightedge and compass, general non-regular convex equilateral pentagons are not constructible using these tools alone, as their vertex positions typically require solving equations beyond quadratic extensions of the rationals.^[20] Concave equilateral pentagons are also possible, featuring at least one reflex interior angle exceeding 180 degrees. A notable example is the sphinx pentagon, a nonconvex shape outlined by assembling six equilateral triangles, yielding five equal exterior sides while resembling the mythical creature in profile.^[9] Self-intersecting equilateral pentagons include the pentagram, a star polygon represented by the Schläfli symbol {5/2}, where five equal-length line segments connect every second vertex of an underlying regular pentagon, forming a five-pointed star with intersecting sides.^[21] For polygons with six or more sides, non-regular equilateral forms become increasingly diverse. Equilateral hexagons, for instance, can manifest as zonogons—centrally symmetric convex polygons with three pairs of parallel sides, all of equal length—allowing variations in the directions of the side pairs while maintaining closure and equal side lengths.^[22] These differ from the regular hexagon by having unequal interior angles, yet they share properties like the ability to tile the plane under specific conditions. Equilateral heptagons with seven equal sides exist as well, though their irregular configurations are more intricate due to the odd number of sides, requiring precise angle adjustments to ensure the polygon closes; for example, convex versions with specified angle sequences have been analyzed in geometric problems.^[23] As the number of sides increases beyond five, equilateral polygons generally relate to edge-transitive symmetries in some cases, where the automorphism group permutes edges uniformly.

Properties

Viviani's Theorem

Viviani's theorem asserts that in an equilateral triangle of side length s, the sum of the perpendicular distances from any interior point to the three sides equals the altitude h = \frac{\sqrt{3}}{2} s.^[24] The theorem generalizes to any convex equilateral polygon. For an n-gon with side length s and area A, the sum of the perpendicular distances d_1 + d_2 + \dots + d_n from any interior point to the sides equals the constant $2A / s.^[25] A proof follows from decomposing the polygon's area A into n triangles, each with base s and height d_i:

A = \sum_{i=1}^n \frac{1}{2} s d_i = \frac{1}{2} s \sum_{i=1}^n d_i,

so \sum d_i = 2A / s.^[26] For tangential equilateral polygons, which admit an incircle with inradius r, the area A = r \cdot (n s / 2), so the constant equals n r. Not all convex equilateral polygons are tangential.^[27]

Relation to Tangential Polygons

A tangential polygon is a polygon that admits an incircle tangent to all its sides.^[28] A necessary condition for a polygon to be tangential is that the sums of the lengths of every other side (alternating sides) are equal; this is necessary and sufficient for quadrilaterals, but for polygons with more than four sides, additional conditions are required.^[29] In the case of an equilateral polygon, where all sides have equal length, this alternating sum condition is automatically satisfied. Consequently, every equilateral quadrilateral, which is a rhombus, possesses an incircle. However, not every equilateral polygon is tangential; for example, certain equilateral hexagons do not admit an incircle unless their vertex angles satisfy specific relations allowing tangency.^[30] For equilateral tangential polygons with an odd number of sides, the configuration is rigid, and the equal side lengths force all tangent segments from the vertices to the points of tangency to be equal in length.^[28] This equality implies that the polygon must also be equiangular, rendering it regular.^[30] Thus, the equilateral triangle, being regular, is tangential, but non-regular equilateral pentagons, for instance, cannot have an incircle. A key property of any tangential polygon is that its inradius r is given by the formula r = \frac{A}{\sigma}, where A is the area and \sigma is the semiperimeter.^[31] For an equilateral tangential polygon with side length s, the semiperimeter \sigma = n s / 2, providing a direct relation between the inradius and the area. This property underpins the expression for the constant in Viviani's theorem as n r for such polygons.^[31]

Measurement and Inequalities

Diagonal Length Constraints

In convex equilateral polygons, the lengths of diagonals are constrained by the triangle inequality applied to the triangles formed during sub-triangulations of the polygon. For a general convex equilateral n-gon with side length a, diagonals can be shorter or longer than a, with the shortest approaching 0 in near-degenerate configurations and the longest approaching nearly \frac{n}{2} a in elongated setups. These bounds arise from considering the polygon as a chain of edges and ensuring convexity, where the triangle inequality in adjacent triangles allows short diagonals when angles are acute and long ones when nearly collinear. For equilateral hexagons specifically, the principal diagonals—those connecting opposite vertices and dividing the hexagon into quadrilaterals—satisfy the existence of one d_1 with \frac{d_1}{a} \leq 2 and one d_2 with \frac{d_2}{a} \geq \sqrt{3}, with equality for the upper bound holding in the regular case. These inequalities ensure that not all principal diagonals exceed twice the side length, while at least one meets the lower bound to maintain closure without excessive contraction. To derive such bounds for specific n, vector geometry can be employed by placing vertices on a coordinate plane with fixed side distances, solving the system |\mathbf{v}_{i+1} - \mathbf{v}_i| = a for i = 1, \dots, n (with \mathbf{v}_{n+1} = \mathbf{v}_1), and optimizing diagonal distances |\mathbf{v}_i - \mathbf{v}_{i+k}| under convexity constraints via Lagrange multipliers or numerical methods. In non-convex or self-intersecting equilateral polygons, these constraints may not hold, as diagonals can become shorter than a due to reflex angles or crossings, or may be undefined in the standard Euclidean sense if intersections alter the interior. For instance, a star-shaped equilateral polygon like a pentagram has intersecting "diagonals" that violate convex bounds, allowing lengths below a relative to the side. Such variations highlight the role of convexity in enforcing the standard inequalities.

Width and Perimeter Relations

The perimeter P of an equilateral n-gon, where each side has length a, is given by the formula P = n a. This relation holds for any equilateral polygon, assuming a fixed side length a, and provides a straightforward measure of the boundary length independent of the interior angles.^[32] The width w of a convex equilateral polygon is defined as the minimal distance between any pair of parallel supporting lines that touch the polygon. In regular polygons, the width varies with direction, and the minimal width of a non-regular equilateral polygon depends on its configuration, reflecting the flexibility in vertex angles while maintaining equal side lengths. For instance, highly elongated shapes exhibit smaller minimal widths compared to more compact ones.^[33] In extremal geometry, significant relations emerge when fixing the minimal width or the diameter. For equilateral convex polygons with fixed minimal width w = 1, the configuration that maximizes the perimeter depends on n: for odd n \geq 3, a trapezoidal shape achieves the maximum, while for even n, the perimeter can be unbounded in certain limiting cases, though practical optima are finite and computed via optimization. This highlights how equilateral constraints influence boundary maximization under width restrictions. When the diameter d (the maximum distance between any two points) is fixed at 1, equilateral convex n-gons maximize their perimeter among equilateral polygons up to specific n, with the optimal perimeter approaching \pi as n grows large, approximating the circle's isodiametric perimeter. For example, non-regular configurations outperform the regular n-gon in maximizing perimeter when n \geq 8 is a power of 2. These bounds provide quantitative insights into how side equality trades off against overall extent.^[32] As a representative example, consider the rhombus, an equilateral quadrilateral with side length a and acute angle \theta. The minimal width is w = a \sin \theta, achieved in the directions perpendicular to the pairs of parallel sides. In other directions making angle \phi with a side, the width can be expressed as w(\phi) = a (|\sin(\phi + \theta)| + |\cos \phi|), illustrating directional variation. The diagonals influence this width by determining the overall span, with longer diagonals generally increasing the average width.

Optimality

Reinhardt Polygons

Reinhardt polygons are convex equilateral n-gons that are optimal solutions to three interrelated geometric optimization problems: achieving the maximal perimeter for a fixed diameter, the maximal minimal width for a fixed diameter, and the maximal minimal width for a fixed perimeter, where n is not a power of 2.^[34] These polygons are inscribed in a Reuleaux polygon of the same diameter, with every vertex of the Reuleaux polygon coinciding with a vertex of the n-gon, ensuring that the extremal properties are realized through specific vertex placements that maximize the relevant metrics.^[34] Named after the German mathematician Karl Reinhardt, who introduced the concept in his seminal 1922 work on extremal polygons of given diameter, these shapes represent the boundary cases in isodiametric problems for equilateral polygons.^[35] Reinhardt's analysis established that regular equilateral polygons achieve optimality only for certain small odd n, such as n=3 and n=5, while for larger n, non-regular configurations are required to attain the maxima.^[36] For n=3, the Reinhardt polygon is the regular equilateral triangle, which inherently satisfies all three optimality conditions due to its symmetry.^[34] For n=4, although 4 is a power of 2 and thus outside the standard definition of Reinhardt polygons, the square serves as the equilateral quadrilateral that maximizes perimeter and width relative to its diameter or perimeter, functioning analogously as an optimal configuration.^[36] For even n greater than 4, constructions of Reinhardt polygons typically involve composing dihedral symmetries with an odd number of factors to form periodic or sporadic patterns that alternate angular orientations, enabling the polygon to span the diameter in multiple directions while maintaining equal side lengths.^[34] These configurations ensure that opposite vertices or edges align to achieve the required extremal widths and perimeters, often resulting in non-regular shapes that outperform regular polygons in the optimization criteria.^[37]

Maximal Configurations

Among equilateral polygons with a fixed perimeter, the regular n-gon achieves the maximal area. This follows from the broader isoperimetric result that, for any n-sided polygon with given perimeter, the regular one encloses the maximum possible area, and the equilateral condition aligns with this optimum since equal side lengths are preserved in the regular configuration.^[38] As the number of sides n increases, the maximal area configuration—the regular equilateral n-gon—approaches the area of a circle with the same perimeter, reflecting the isoperimetric inequality's limit where the circle is optimal among all plane figures.^[36] For equilateral polygons with a fixed diameter (the maximum distance between any two points), the regular n-gon again maximizes the area among all such configurations. In the case of n=5, the optimal equilateral pentagon is regular, featuring uniform interior angles of 108° each, which distributes the angles evenly to enclose the largest area without violating the diameter constraint. Among equilateral n-gons with fixed diameter, the regular configuration maximizes the perimeter only for n=3 and n=5. For n > 5, non-regular equilateral n-gons achieve larger perimeters than the regular n-gon. Examples of such extremal equilateral configurations include certain Reinhardt polygons. For n > 5, non-equilateral n-gons can also surpass the maximum perimeter achieved by equilateral polygons.

References

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