GitHub

Polygons

Introduction

A polygon is a planar (two-dimensional) geometric figure which is also called an N-corner, because their number of corners is N – a natural number greater than

  1. To construct a polygon, start with N points in the plane and connect them with N lines in such a way that:

  2. they do not intersect, and

  3. the starting point of the first line coincides with the ending point of the last line

These two properties guarantee that such a closed polygon (all lines) is actually the boundary of a two-dimensional figure – the N-corner.

The line segments are called the sides or edges of the polygon, and the points where the sides meet are called the vertices or corners. Polygons can have various numbers of sides, and they are classified based on the number of sides they possess.

Here are some common types of polygons based on the number of sides (the first two we have described in detail in previous sections):

  • Triangle (3 sides): The simplest polygon, with three edges and three vertices.
  • Quadrilateral (4 sides): A polygon with four edges and four vertices. Examples include squares, rectangles, and trapezoids.
  • Pentagon (5 sides): A polygon with five edges and five vertices.
  • Hexagon (6 sides): A polygon with six edges and six vertices.
  • Heptagon (7 sides): A polygon with seven edges and seven vertices.
  • Octagon (8 sides): A polygon with eight edges and eight vertices.
  • Nonagon (9 sides): A polygon with nine edges and nine vertices.
  • Decagon (10 sides): A polygon with ten edges and ten vertices.
  • Hendecagon (11 sides): A polygon with eleven edges and eleven vertices.
  • Dodecagon (12 sides): A polygon with twelve edges and twelve vertices.

Classification of Polygons

Polygons are typically classified into two main categories: convex and concave.

  1. Convex Polygons: A polygon is convex if all its interior angles are less than 180 degrees, and no line segment between any two points on its boundary goes outside the polygon. In simpler terms, a convex polygon "bulges" outward.

    A more exact definition is that a convex polygon $\Lambda$ with two internal points $X$, $Y$, has the property that every point $Z$ of the connecting line between $X$ and $Y$ lies inside $\Lambda$. Mathematically, all points $Z$ of the form: $Z = tX + (1-t)Y \text{ for } t \in [0, 1]$ lie inside $\Lambda$.

  2. Concave Polygons: A polygon is concave if at least one of its interior angles is greater than 180 degrees. This means that there is at least one line segment between two points on its boundary that goes outside the polygon, causing it to "cave" inward.

    In concave polygons $\Lambda$, internal points $X$ and $Y$ exist in $\Lambda$ such that some point(s) $Z$ on their connecting line is not in $\Lambda$.

Convex vs Concave Polygons X Y Convex Polygon All interior angles < 180° Line XY stays inside X Y θ > 180° Concave Polygon At least one interior angle > 180° Line XY goes outside Convex Concave Line inside polygon Line outside polygon

Regular vs. Irregular Polygons

Polygons can also be classified as regular or irregular:

  1. Regular Polygons: A regular polygon has all sides of equal length and all interior angles of equal measure. Examples include equilateral triangles and squares.

  2. Irregular Polygons: An irregular polygon does not have equal side lengths or equal angles. Most polygons encountered in everyday life are irregular.

Regular vs Irregular Polygons s s s s s s 120° 120° 120° Regular Hexagon All sides equal: s All angles equal: 120° a b c d e f α β γ δ ε ζ Irregular Hexagon Different side lengths: a,b,c,d,e,f Different angles: α,β,γ,δ,ε,ζ Regular (equal sides & angles) Irregular (unequal sides & angles)

Properties of Polygons

Polygons have several important properties that are useful in geometry:

  1. Sum of Interior Angles: The sum of the interior angles of a polygon with N sides is given by the formula: $\text{Sum of interior angles} = (N - 2) \times 180^\circ$. For example, a triangle (3 sides) has a sum of interior angles equal to $180^\circ$, while a quadrilateral (4 sides) has a sum of $360^\circ$.
  2. Exterior Angles: The sum of the exterior angles of any polygon is always $360^\circ$, regardless of the number of sides. Each exterior angle is formed by extending one side of the polygon at a vertex. For a regular polygon, each exterior angle can be calculated as: $\text{Exterior angle} = \frac{360^\circ}{N}$
  3. Diagonals: A diagonal is a line segment connecting two non-adjacent vertices of a polygon. The number of diagonals (D) in a polygon with N sides can be calculated using the formula: $D = \frac{N(N - 3)}{2}$. For example, a pentagon (5 sides) has 5 diagonals, while a hexagon (6 sides) has 9 diagonals.
  4. Area: The area of a polygon can be calculated using various methods depending on the type of polygon. For regular polygons, the area can be calculated using the formula: $\text{Area} = \frac{1}{4} N s^2 \cot\left(\frac{\pi}{N}\right)$ where $s$ is the length of a side. If you circumscribe the regular polygon with a circle of radius $r$ (the distance from the center to a vertex), the area can also be expressed as: $\text{Area} = \frac{1}{2} N r^2 \sin\left(\frac{2\pi}{N}\right)$. For irregular polygons, methods such as triangulation or the shoelace formula can be used.
  5. Perimeter: The perimeter of a polygon is the total length of its sides. For a polygon with N sides, the perimeter $P$ can be calculated as: $P = s_1 + s_2 + s_3 + ... + s_N$ where $s_1, s_2, ..., s_N$ are the lengths of the sides.
  6. Symmetry: Regular polygons exhibit a high degree of symmetry. They have multiple lines of symmetry and rotational symmetry. The number of lines of symmetry in a regular polygon is equal to the number of sides $N$.

Triangulation

Triangulation is the process of dividing any polygon into triangles by drawing non-intersecting diagonals. This technique is fundamental in geometry because triangles are the simplest polygons, and we already know how to calculate their properties (area, angles, side lengths) using basic geometric formulas.

Why Triangulation Works

The number of triangles formed when triangulating a polygon with $N$ sides is always: $\text{Number of triangles} = N - 2$

This formula works because:

  • Each triangle contributes $180°$ to the total angle sum
  • The total interior angle sum of an $N$-sided polygon is $(N-2) \times 180°$
  • Therefore: $(N-2) \text{ triangles} \times 180° = (N-2) \times 180°$

Basic Triangulation Methods

1. Fan Triangulation The simplest method is to choose one vertex and connect it to all non-adjacent vertices. This creates $(N-2)$ triangles that all share the chosen vertex as a common point.

Example - Pentagon Fan Triangulation:

  • Start with pentagon $ABCDE$
  • Choose vertex $A$ as the fan center
  • Draw diagonals: $AC$ and $AD$
  • Result: 3 triangles ($\triangle ABC$, $\triangle ACD$, $\triangle ADE$)
Pentagon Fan Triangulation A B C D E △ABC △ACD △ADE AC AD Fan Triangulation Steps: 1. Choose vertex A as the fan center 2. Draw diagonals AC and AD from vertex A 3. Result: N-2 = 5-2 = 3 triangles Verification: Pentagon: N = 5 sides Triangles = N - 2 = 5 - 2 = 3 ✓ Angles: 3 × 180° = 540° ✓ Triangle ABC Triangle ACD Triangle ADE Diagonals

Example - Hexagon Fan Triangulation:

  • Start with hexagon $ABCDEF$
  • Choose vertex $A$ as the fan center
  • Draw diagonals: $AC$, $AD$, and $AE$
  • Result: 4 triangles ($\triangle ABC$, $\triangle ACD$, $\triangle ADE$, $\triangle AEF$)
Hexagon Fan Triangulation A B C D E F △ABC △ACD △ADE △AEF AC AD AE Fan Triangulation Steps: 1. Choose vertex A as the fan center 2. Draw diagonals AC, AD, and AE from vertex A 3. Result: N-2 = 6-2 = 4 triangles Verification: Hexagon: N = 6 sides Triangles = N - 2 = 6 - 2 = 4 ✓ Angles: 4 × 180° = 720° ✓ △ABC △ACD △ADE △AEF Diagonals from A Note: All triangles share vertex A as a common point. This creates a "fan" pattern radiating from A

2. Alternative Triangulation Methods The same polygon can be triangulated in multiple ways. For example, a quadrilateral $ABCD$ can be triangulated as:

  • Method 1: Diagonal $AC$ creates $\triangle ABC$ and $\triangle ACD$
  • Method 2: Diagonal $BD$ creates $\triangle ABD$ and $\triangle BCD$

Both methods create exactly 2 triangles, as predicted by the formula $N-2 = 4-2 = 2$.

Alternative Triangulation Methods A B C D △ABC △ACD AC Method 1: Diagonal AC A B C D △ABD △BCD BD Method 2: Diagonal BD Both Methods Create Exactly 2 Triangles Formula: N - 2 = 4 - 2 = 2 triangles ✓ Total angles: 2 × 180° = 360° ✓ Method 1 Triangles: △ABC △ACD Method 2 Triangles: △ABD △BCD Diagonal AC Diagonal BD

Using Triangulation to Calculate Area

Triangulation is particularly useful for finding the area of irregular polygons:

Step-by-Step Process:

  1. Triangulate the polygon using any valid method
  2. Measure or calculate the side lengths of each triangle
  3. Apply triangle area formulas to each triangle:
    • Use the basic formula: Area = $\frac{1}{2} \times \text{base} \times \text{height}$
    • Or use Heron's formula for triangles with known side lengths
    • Or use trigonometric formula: Area = $\frac{1}{2}ab\sin(C)$
  4. Sum all triangle areas to get the total polygon area

Example - Pentagon Area Calculation: Given pentagon with vertices and using fan triangulation from vertex $A$:

  • Triangle 1 area: $A_1$ (calculated using appropriate triangle formula)
  • Triangle 2 area: $A_2$
  • Triangle 3 area: $A_3$
  • Total pentagon area = $A_1 + A_2 + A_3$

Verification Methods

To verify your triangulation is correct:

  1. Count triangles: Should equal $N-2$
  2. Check angle sum: All triangle angles should sum to $(N-2) \times 180°$
  3. Verify no overlaps: Triangles should not overlap each other
  4. Confirm coverage: Triangles should cover the entire polygon interior

Practical Applications

1. Area Calculation

  • Breaking complex shapes into manageable triangular pieces
  • Each triangle can be solved using familiar formulas
  • Particularly useful for irregular polygons where direct area formulas don't exist

2. Construction and Design

  • Architectural applications where complex shapes need to be built from simpler triangular components
  • Engineering applications requiring structural analysis of polygonal shapes

3. Problem Solving

  • Converting complex polygon problems into simpler triangle problems
  • Using known triangle properties to solve unknown polygon properties

The power of triangulation lies in its ability to reduce complex polygonal problems to collections of simple triangle problems, using only the basic geometric tools we've already developed.

Cyclic Quadrilaterals

A quadrilateral can be inscribed in a circle (cyclic quadrilateral) if and only if the sum of each pair of opposite angles is $180^\circ$. This property is useful in various geometric proofs and calculations.

Cyclic Quadrilateral Inscribed in a Circle A B C D ∠A ∠B ∠C ∠D Property: ∠A + ∠C = 180° and ∠B + ∠D = 180°

Calculating Area of a Cyclic Quadrilateral

The area of a cyclic quadrilateral can be calculated using Brahmagupta's formula, which is an extension of Heron's formula for triangles. If a cyclic quadrilateral has side lengths $a$, $b$, $c$, and $d$, the area $K$ can be calculated as follows: $K = \sqrt{(s-a)(s-b)(s-c)(s-d)}$ where $s$ is the semiperimeter: $s = \frac{a + b + c + d}{2}$ This formula is particularly useful for finding the area when the side lengths are known, without needing to know the angles or the height of the quadrilateral.