polygons


image: triangle with angles marked a, b, c and d
A polygon is a 2-dimensional closed shape with straight sides. In this section we will revise the properties of polygons.

Angle properties of triangles

We already know that the angles in a triangle add up to 180°. Angles on a straight line also add up to 180°.
image: triangle with angles marked a, b, c and d
In the diagram, we see that:
a + b + c = 180° (angles in a triangle)
and
c + d = 180° (angles on a straight line).
If we rearrange both equations (subtract c from both sides), we get:
a + b = 180° - c and d = 180° - c.
Therefore, a + b and d must be the same (they are both equal to 180° - c):
a + b = d
Now look again at the diagram.
image: triangle with angles marked a, b, c and d
The exterior angle is equal to the sum of the two opposite interior angles.
This is true for any triangle.
Question
Find each angle marked with a letter, giving reasons for your answer.


Quadrilaterals

The angles in a quadrilateral add up to 360º. There are other facts that you will need to know about special types of quadrilaterals. These are shown below:

Square

image: square, quadrilateral
A square is a regular quadrilateral. All of its angles are equal (90°). All of its sides are of equal length. Opposite sides are parallel. The diagonals bisect each other at 90°. The diagonals are equal in length. It has 4 lines of symmetry. Order of rotational symmetry: 4.

Rhombus

image: rhombus, quadrilateral
Diagonally opposite angles are equal. All of its sides are of equal length. Opposite sides are parallel. Diagonals bisect each other at 90°. It has 2 lines of symmetry. Order of rotational symmetry: 2.

Rectangle

image: rectangle, quadrilateral
All angles are equal (90°). Opposite sides are of equal length. Opposite sides are parallel. The diagonals bisect each other. The diagonals are equal in length. It has 2 lines of symmetry. Order of rotational symmetry: 2.

Parallelogram

Diagonally opposite angles are equal. Opposite sides are of equal length. Opposite sides are parallel. The diagonals bisect each other. It has no lines of symmetry. Order of rotational symmetry: 2.

Trapezium

image: trapezium, quadrilateral
One pair of opposite sides is parallel. It has no lines of symmetry. It has no rotational symmetry.

Kite

image: kite, quadrilateral
Two pairs of sides are of equal length. One pair of diagonally opposite angles is equal. Only one diagonal is bisected by the other. The diagonals cross at 90°. It has 1 line of symmetry. It has no rotational symmetry.
Do not be too alarmed by all these facts - you probably know most of them already!
image: triange with angles a and b labelled, one angle given as 70 degrees and an external angle in relation to b is given as 120 degreeshfns

Regular and irregular polygons

The simplest polygon is a triangle (a 3-sided shape). Polygons of all types can be regular or irregular.
A regular polygon has sides of equal length, and all its interior angles are of equal size.
Irregular polygons can have sides of any length and angles of any size.
Here are the names of some common polygons:

 

Number of sidesName of polygonShape
3triangle
image: triangle
4quadrilateral
image: quadrilateral
5pentagon
image: pentagon
6hexagon
image: hexagon
8octagon
image: octagon
10decagon

Angle properties of polygons

In your exam, you might be asked to find angles of polygons.
The formula for calculating the sum of the interior angles of a regular polygon is: (n - 2) × 180° where n is the number of sides of the polygon.
This formula comes from dividing the polygon up into triangles using full diagonals.
We already know that the interior angles of a triangle add up to 180°. For any polygon, count up how many triangles it can be split into. Then multiply the number of triangles by 180.
image: quadrilateral
This quadrilateral has been divided into two triangles, so the interior angles add up to 2 × 180 = 360°.
image: pentagon
This pentagon has been divided into three triangles, so the interior angles add up to 3 × 180 = 540°.
In the same way, a hexagon can be divided into 4 triangles, a 7-sided polygon into 5 triangles etc.
Can you see the pattern forming? The number of triangles is equal to the number of sides minus 2.
Question
What is the sum of the interior angles of an octagon?
toggle answer
If you find it difficult to remember formulae, just add 180° each time in the following way:

 

Number of sidesSum of interior anglesShape
3180°
image: triangle
4180° + 180° = 360°
image: quadrilateral
5360° + 180° = 540°
image: pentagon
6540° + 180° = 720°
image: hexagon
8900° + 180° = 1080°
image: octagon
101260° + 180° = 1440°
image: decagon
The exterior angle of a polygon and its corresponding interior angle always add up to 180° (because they make a straight line).
image: interior and exterior angles of a pentagon
For any polygon, the sum of its exterior angles is 360°.
You can see this because if you imagine 'walking' all the way round the outside of a polygon you make o

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