Polygons and Regular Polygons
A polygon is a closed plane figure bounded by straight line segments, and a regular polygon has all sides equal and all angles equal. In brief: a polygon with n sides has n vertices and n angles. A regular hexagon has six equal sides, each interior angle is 120° and its area with side a is (3√3/2)a2. Polygons are named by the number of sides: triangle, quadrilateral, pentagon, hexagon, heptagon, octagon and so on.
Polygons describe the outlines of fields, buildings, tiles and logos. Regular polygons, with their high symmetry, are special: they can be inscribed in a circle, and they make up many tilings and designs. This guide explains the terminology, the types of polygons, the formulas for regular polygons (angles, area, perimeter) and shows which regular polygons tile the plane.
What is a polygon?
A polygon consists of line segments (sides) joined end to end, forming a closed figure that does not cross itself (a simple polygon). The points where the sides meet are vertices. The word comes from the Greek for “many angles”. A polygon has as many vertices as sides. A segment joining two non-adjacent vertices is a diagonal.
The names follow the number of sides n: 3 triangle, 4 quadrilateral, 5 pentagon, 6 hexagon, 7 heptagon, 8 octagon, 9 nonagon, 10 decagon, 11 hendecagon, 12 dodecagon. Beyond that, one says “n-gon”.
What types of polygons are there?
- Convex: every interior angle is less than 180°, and any segment between two points of the polygon lies inside it.
- Concave: at least one interior angle is greater than 180°, like an arrowhead.
- Equilateral: all sides are equal (a rhombus is equilateral but not necessarily equiangular).
- Equiangular: all angles are equal (a rectangle is equiangular but not necessarily equilateral).
- Regular: both equilateral and equiangular.
What is a regular polygon?
A regular polygon has equal sides and equal angles. For n sides, each interior angle is (n − 2) · 180°/n and each exterior angle is 360°/n. Every regular polygon has a circumscribed circle through the vertices and an inscribed circle touching the sides, with a common center O, and it has n axes of symmetry and rotational symmetry of order n. The center divides it into n congruent isosceles triangles with apex angle 360°/n.
What are the formulas for regular polygons?
For a regular n-gon with side a, circumradius R and inradius (apothem) r:
- Perimeter: P = na.
- Apothem: r = a/(2 tan(180°/n)).
- Circumradius: R = a/(2 sin(180°/n)).
- Area: A = (1/2) · P · r = (n a2)/(4 tan(180°/n)).
Example 1: regular hexagon with side 6
The hexagon consists of 6 equilateral triangles with side 6. The area of each is (√3/4) · 36 = 9√3, so A = 54√3 ≈ 93.5. The circumradius is 6 and the apothem is 3√3 ≈ 5.20. Check with the formula: A = (1/2) · 36 · 5.196 = 93.5 ✓.
Example 2: regular octagon with side 4
Apothem r = 4/(2 tan 22.5°) = 4/(2 · 0.41421) = 4.828. Area = (1/2) · 32 · 4.828 = 77.25. The formula A = 2(1 + √2)a2 gives 2 · 2.4142 · 16 = 77.25 ✓.
Example 3: pentagon from the circumradius
A regular pentagon with R = 5: side a = 2R sin 36° = 10 · 0.5878 = 5.878, apothem r = R cos 36° = 5 · 0.8090 = 4.045, area = (1/2) · 5 · 5.878 · 4.045 = 59.44.
Regular polygons from 3 to 12 sides
| Sides n | Name | Interior angle | Exterior angle | Area for side 1 |
|---|---|---|---|---|
| 3 | triangle | 60° | 120° | 0.4330 |
| 4 | square | 90° | 90° | 1.0000 |
| 5 | pentagon | 108° | 72° | 1.7205 |
| 6 | hexagon | 120° | 60° | 2.5981 |
| 7 | heptagon | about 128.57° | about 51.43° | 3.6339 |
| 8 | octagon | 135° | 45° | 4.8284 |
| 9 | nonagon | 140° | 40° | 6.1818 |
| 10 | decagon | 144° | 36° | 7.6942 |
| 11 | hendecagon | about 147.27° | about 32.73° | 9.3656 |
| 12 | dodecagon | 150° | 30° | 11.1962 |
To get the area for another side length a, multiply the table value by a2. For example, a regular decagon with side 3 has area 7.6942 · 9 ≈ 69.25.
Which regular polygons tile the plane?
Only three regular polygons tile the plane by themselves without gaps: the equilateral triangle, the square and the regular hexagon. The reason is that the interior angle must divide 360°: 360°/60° = 6, 360°/90° = 4, 360°/120° = 3. For the pentagon, 360°/108° is not a whole number, so copies leave gaps or overlap. Combinations of different regular polygons, such as squares with octagons (the “4.8.8” pattern) and triangles with hexagons, also tile the plane. Honeycombs use hexagons, since they enclose a given area with the least perimeter among the three.
Where do polygons come from?
Regular polygons were studied by the Greeks, and Euclid’s Elements contains constructions of the equilateral triangle, the square, the pentagon, the hexagon and the 15-gon. Archimedes estimated π by using regular polygons with 96 sides inscribed in and circumscribed about a circle, which gave 3 + 10/71 < π < 3 + 1/7. Which regular polygons can be constructed with a compass and straightedge was settled by Gauss and Wantzel.
Where are they used today?
Nuts and bolts are hexagonal for wrench grip, stop signs are octagons, and many coins are polygons. Honeycombs and the structure of graphene are hexagonal. Computer graphics approximate curved surfaces with polygons, and the more sides a polygon has, the closer it looks to a circle: a regular 100-gon is nearly indistinguishable from its circumscribed circle.
Which mistakes are common?
Assuming that equal sides imply equal angles
A rhombus has equal sides, but unequal angles in general. A regular polygon needs both.
Mixing up interior and exterior angles
The interior angle plus the exterior angle at a vertex equals 180°. The exterior angle is 360°/n for regular polygons.
Using the formula for regular polygons for irregular ones
The apothem formula works only for regular polygons. Irregular polygons are split into triangles.
Forgetting the diagonals
The number of diagonals is n(n − 3)/2, not n.