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Year 10+ Plane Geometry


Polygons

Polygons

A polygon is any plane (flat) shape with straight edges.

How Do You Tell if It's a Polygon?

There are two things to check:

  1. All the edges are straight
  2. The shape is closed. That is, there aren't any gaps as the final edge ends where the first edge begins.
 Polygons

Top Secret Polygon Terminology

Well, maybe not. But here are some terms that are used to describe polygons.

Regular and Irregular

In a regular polygon, all the edges have the same length and all the angles have the same size. In an irregular polygon, they don't.

 Polygons

Concave and Convex

A convex polygon is one that doesn't have any dents in it: none of its interior angles are bigger than \(180^\circ\). A convex polygon looks a bit like a cave: it has at least one interior angle that is bigger than \(180^\circ\), so it has a dent in it.

Another way of thinking of this is that any line segment joining any two points inside a convex polygon stays entirely inside the polygon. You can find two points of a concave polygon that can't be joined by a line segment that stays entirely inside the polygon. See if you can find two points like this in the concave polygon in the picture below.

 Polygons

Simple and Complex

In a simple polygon, the edges only meet at the vertices (corner points). In a complex polygon, the edges can cross at points other than the vertices. Mathematicians say that a complex polygon intersects itself.

Lots of facts about polygons only work for simple polygons, so be careful about the facts you try to apply to complex polygons. In the pictures, the simple polygon is a pentagon (has 5 edges), and the complex polygon is a hexagon (has 6 edges).

 Polygons

Star Polygons

Enough said, really. These are all complex pentagons. Here are a few of them:

 Polygons

Naming Polygons

There's a complex system of naming polygons available using all sorts of Greek prefixes. For example, you could call a 97-sided polygon a enneacontaheptagon if you really wanted to. However, once we get past about 12 sides, it really makes more sense to refer to a polygon with \(n\) sides as an \(n\)-gon. Let's have a look at the names of some common polygons, and the sizes of their interior angles.

Name of Polygon Number of Sides Picture Interior Angles if Regular
Triangle 3  Polygons \(60^\circ\)
Quadrilateral 4  Polygons \(90^\circ\)
Pentagon 5  Polygons \(108^\circ\)
Hexagon 6  Polygons \(120^\circ\)
Heptagon (or Septagon) 7  Polygons \(128.57\dots^\circ\)
Octagon 8  Polygons \(135^\circ\)
Nonagon 9  Polygons \(140^\circ\)
Decagon 10  Polygons \(144^\circ\)
Hendecagon (or undecagon) 11 \(147.27^\circ\)
Dodecagon 12 \(150^\circ\) Icosagon 20 \(162^\circ\)
30-gon 30 \(168^\circ\)
90-gon 90 \(176^\circ\)
1,000,000-gon 1,000,000 \(180^\circ\) (about)
n-gon n \(\dfrac{n}{n-2} \times 180^\circ\)

As the number of sides of a regular polygon gets larger, the interior angles get closer and closer to \(180^\circ\).

Polygons in Nature and Architecture

 Polygons

How many triangles can you spot in this stained glass ceiling? This ceiling is part of the National Gallery of Victoria in Melbourne.


 Polygons

The faces of salt crystals are shaped like quadrilaterals (squares).


 Polygons

Bees build the cells of their honeycomb in the shape of hexagons.


 Polygons

This, very originally named, building at the University of Heidelberg is called the Octagon Building.


Have a look around you and see if you can spot any more examples of polygons in nature, art or architecture. Have fun!

Description

In these chapters you will learn about plane geometry topics such as 

  • Area (Irregular polygons, plane shapes etc)
  • Perimeter
  • Conic sections (Circle, Ellipse, Hyperbola etc)
  • Polygons (Congruent, polygons, similar, triangles etc)
  • Transformations and symmetry (Reflection, symmetry, transformations etc)

etc

Even though these chapters are marked for Year 10 or higher students, several topics are for students in Year 8 or higher



Audience

Year 10 or higher, suitable for Year 8 + students as well.

Learning Objectives

Learn about Plane Geometry

Author: Subject Coach
Added on: 28th Sep 2018

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