Looking at Polygons
Introduction

Most of the shapes that you are familiar with and that you see every day are polygons.


The word itself means 'many angles', and every polygon has the same number of angles as sides.
What is a polygon?
Distinguishing polygons from other shapes by eye is relatively simple.
Figure 1.   Polygons.
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Figure 2.   Non-polygons.
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You probably understand the distinction between polygons and other shapes just from looking at the diagrams above. In precise terms, though, the definition of a polygon is:


A closed plane figure bounded by straight line segments.



Look at the diagrams below and then answer the questions.

Figure 3.   Some shapes.
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Which of the shapes in Fig.3 is a
polygon
A polygon is a closed, planar figure bounded by straight line segments.
polygon
?
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Which of the shapes in Fig.3 is not a polygon because it is not
closed
A closed curve is one that is continuous and that begins and ends in the same place.
closed
and bounded?
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Which of the shapes in Fig.3 is not a polygon because it is not bounded solely by straight lines?
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Polygons can be classified according to the number of sides they have. Look at the table below and use it to answer the questions that follow.


Name of polygon Number of sides
Triangle 3
Quadrilateral 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
Nonagon 9
Decagon 10
Undecagon 11
Dodecagon 12

Look at the shapes below and match each to its name.
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List as many different types of quadrilaterals as you can.

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Describing polygons
The name of a polygon indicates only the number of sides it has, and there are many different shapes that have the same number of sides. So to describe a polygon accurately we need to look at other aspects of its shape. To start with, it is useful to label the vertices and sides. The vertices of a shape are often named in sequence either clockwise or anticlockwise with letters from the alphabet.

Figure 4.   Labelling a polygon's sides and angles.
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To help us reason about
angle
An angle is a measure of turning. Angles are measured in degrees. The symbol for an angle is .
angle
and line relationships in polygons, we often talk about sides, angles or vertices being adjacent (next to one another). So in Fig.4 above:
  • side AB is adjacent to side BC

  • vertex D is adjacent to vertex E

  • angle x is adjacent to angle y



Two adjacent sides of a polygon create an interior angle. In Fig.5 below, the interior angles of three polygons have been marked.

Figure 5.   Interior angles in polygons.
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If all of a polygon's interior angles are less than 180, it is called a convex polygon. If one or more of its angles are
reflex
A reflex angle is over 180 but less than 360.
reflex
angles (over 180), it is described as concave.

Figure 6.   A concave and a convex pentagon.
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Describe each shape below as either '
concave
A polygon is concave if one or more of its interior angles is greater than 180.
concave
' or '
convex
A polygon is convex if all of its interior angles are less than 180.
convex
'.
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We also refer to the exterior angles of a convex polygon. These are the angles formed by a side and an adjacent side that has been extended, as illustrated below.

Figure 7.   Exterior angles of a pentagon.
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We can
produce
When a line segment is produced it is extended in the same direction.
produce
(extend) each of the other sides of the above irregular
pentagon
A pentagon is a five-sided polygon.
pentagon
. Press the button to see this. Fill in the values of the exterior and interior values at each
vertex

A vertex is defined as the common endpoint of two lines.

vertex
in the table below.

For each vertex, add together the interior and the exterior angles.


Vertex A B C D E
Exterior angle
Interior angle
Total




What you should be able to see is that the interior and exterior angles at every vertex of this polygon sum to 180.

For any convex polygon, do you think the interior and exterior angles will sum to 180°?
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This is because...
  • This is true, but it does not explain why the interior and exterior angles at a vertex sum to 180.
  • This is true, but it does not explain why the interior and exterior angles at a vertex sum to 180.
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The sentences below describe the shapes to their left. Complete the sentences by filling in the first blank with either 'interior' or 'exterior' and the second blank with the value of the labelled angle.


  • Angle a in this rectangle is an angle equal to .
         
    Angle b in this hexagon is an angle equal to .
         
    Angle c in this octagon is an angle equal to .

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What is a rectangle?
You will already recognize what a rectangle looks like, but how would you describe it to someone who does not recognize the name you use for it?


Click on the figure below to interact with the model.

 Figure 8.  A rectangle.



You could start by describing how many sides it has.

Complete the sentence below.

  • A rectangle is a polygon with sides so it is known as a .
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We have already established that there are plenty of shapes with the same number of sides as a rectangle that are not rectangular, so we need to supply more information about it to define it adequately. We know that:

'Every rectangle has two sets of
parallel
Two lines, curves or planes are said to be parallel if the perpendicular distance between them is always the same.
parallel
sides.'

With this extra information have we fully described a rectangle?
  • Wrong. A parallelogram also has two sets of parallel lines, but it is not a rectangle.
  • That's right! A parallelogram also has two sets of parallel lines, but it is not a rectangle.
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We also know that:

'The four interior angles of a rectangle all measure 90.'

Is there a
quadrilateral
A quadrilateral is a polygon with four sides.
quadrilateral
whose interior angles are all 90, and which has two sets of parallel lines that is not a rectangle?
  • In fact there is not. A square is actually a type of rectangle.
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So we have defined a rectangle. In fact, we don't need to mention the existence of its parallel sides, since any quadrilateral with four interior angles of 90 will have two sets of parallel sides. So our definition can be shortened:

A rectangle is a quadrilateral with four interior angles of 90.



Interestingly, a square is actually a special type of rectangle: it is a rectangle with four sides of equal length.


Click on the figure below to interact with the model.

 Figure 9.  A square.



What is a parallelogram?
A
parallelogram
A parallelogram is a quadrilateral with two sets of parallel sides.
parallelogram
is a quadrilateral that has two sets of parallel sides. In fact, this is all that is need to define a parallelogram. Parallelograms can often be found in mechanical designs, such as in Fig.10.

Figure 10.   The steering mechanism of a go-kart.
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Adjust the shape of the parallelogram below by dragging its corners and sides. Notice how its angles are related.


Click on the figure below to interact with the model.

 Figure 11.  A parallelogram.



Opposite sides in a parallelogram are always parallel, which ensures that there are two pairs of sides made up of lines of the same length. Drag on one side of the parallelogram in Fig.11 to alter the internal angles.

Set the value of an
interior angle
An interior angle is the angle between adjacent sides at a vertex of a polygon.
interior angle
in Fig.11 to 90. Which of the following words describes a shape which cannot be created?
  • Click here to mark the question


Since a parallelogram is not defined in terms of the size of its angles, it may have four angles of 90. A
rectangle
A rectangle is a quadrilateral with four interior angles of 90.
rectangle
is, therefore, a type of parallelogram.

And just as a square is a rectangle with equal-length sides, so a rhombus is a parallelogram with equal-length sides.


Click on the figure below to interact with the model.

 Figure 12.  A rhombus.

'Every square is a rhombus.' True or false?
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'Every rhombus is a square.' True or false?
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Can a parallelogram be constructed from two identical triangles? (Use the two congruent triangles in the model in Fig.13 below to test your answer. Move the triangles towards each other and try to construct a parallelogram.)
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Click on the figure below to interact with the model.

 Figure 13.  Making a parallelogram.

What is a trapezium?
A
trapezium
A trapezium is a quadrilateral with one set of parallel sides and one set of non-parallel sides.
trapezium
is a quadrilateral with one set of parallel lines and one set of non-parallel lines. All the shapes in Fig.14 below are trapezia.

Figure 14.   Trapezia.
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Investigate the properties of trapezia with the shape below in Fig.15.


Click on the figure below to interact with the model.

 Figure 15.  A trapezium.

Look at the shapes below and select the correct word to describe them.
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Regular polygons
Regular polygons have equal sides and equal interior angles. All the shapes below are regular polygons.
Figure 16.   A selection of regular polygons.
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Complete the following sentences.

  • All regular polygons are . Each regular polygon's interior angles are equal, so their exterior angles are .
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Fig.17 below shows a
regular polygon
A regular polygon has sides of equal length and interior angles of the same size.
regular polygon
. Connect each vertex to the centre by clicking on the button next to the shape.


Click on the figure below to interact with the model.

 Figure 17.  A polygon.

What do you notice about the triangles created in the polygon of Fig.17? (Tick all those that apply.)
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So a regular polygon with n sides can be thought of as a pattern of n congruent isosceles triangles.

In Fig.18 below, move the triangles so that they are arranged around a
point
  • A point has no properties except position. It is an object with zero dimensions.
  • Points in the x-y plane can be specified using x and y coordinates.
point
. What type of regular polygon do they make?

  • It is a .
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Click on the figure below to interact with the model.

 Figure 18.  Constructing a polygon using equilateral triangles.

Summary

Polygons are closed figures bounded by straight line segments.

Polygons are often classified according to how many sides they have (e.g. pentagons, hexagons, heptagons).

Every polygon is either concave or convex.

For every interior angle in a convex polygon, there is a corresponding exterior angle.

Regular polygons have sides of equal length and interior angles of equal size.
Exercises
1. Which of the shapes in Fig.19 below are polygons? (Tick as many as apply.)
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Figure 19.   Assorted shapes.
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2. Match each of the polygons in Fig.19 above to its description.
  • Shape 1
    Shape 2
    Shape 3
    Shape 4
    Shape 5
    Shape 6
    Shape 7
    Shape 8
    Shape 9
    Shape 10
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3. Look at the shapes in Fig.20 below, then complete the following sentence.

  • All the coloured shapes in Fig.20 below are polygons, since the sides of each polygon are all the same length and each one has interior angles that are all the same size.
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Figure 20.   Five polygons.
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4. Measure and record the interior angle for each of the shapes in Fig.20 above.

  • Pentagon 
    Hexagon 
    Octagon 
    Nonagon 
    Decagon 

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5. As shown in Fig.21 below, the structure of a skateboard is the same shape as a type of quadrilateral. What type is it?

  • It is a(n) .
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Figure 21.   A skateboard.
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6. The pattern of lines in Fig.22 below contains many shapes. Match each shape to its name.
  • Shape ABC
    Shape ABDF
    Shape ACEF
    Shape ABEF
    Shape ACDF
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Figure 22.   Several lines.
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7. In Fig.22 above, the angle BAC is 38. What is the interior angle ACE of the parallelogram ACEF?
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8. Look at the skip in Fig.23 below, then complete this sentence.

  • The shape of the skip is a and the angle s is .
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Figure 23.   A rubbish skip.
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