Polygons

A 'polygon' is the mathematical term used to describe a two-dimensional (2D) closed shape with straight sides. The term 'polygon' is Greek; with 'poly' meaning 'many' and 'gon' meaning angle.

Polygons can either be regular or irregular. If all of the angles of a 2D shape are equal and all the sides are equal, then it is regular polygon. If the angles are not equal and the sides are not equal, then the shape is an irregular polygon.

You can also have concave or convex polygons. A convex polygon is a 2D shape which has no angles pointing inwards. With a convex polygon, none of the internal angles can be more than 180°. Alternatively, if any internal angle is more than 180°, then the polygon is concave.

There are also simple and complex polygons. A simple polygon only has one boundary, and does not cross over itself. Conversely, a complex polygon will intersect itself and therefore has more than one boundary. For example:

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Key Concepts

In the new linear GCSE Maths paper, you will be required to solve various mathematical problems involving polygons. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:

  • Calculate and use the sums of the interior and exterior angles of polygons
  • Recognise the properties and definitions of special types of quadrilaterals including; a square, rectangle, parallelogram, trapezium, kite and rhombus
  • Understand and use the angle properties of lines, triangles, and quadrilaterals

Listed below are a series of summaries and worked examples to help you solidify your knowledge about polygons.

Worked Examples

1 - Angle properties of triangles
A triangle is a 3-sided polygon. The angles within a triangle add up to 180°.

Angles on a straight line also add up to 180°. As a result, there is a rule which applies to all triangles:

The exterior angle is equal to the sum of the two opposite interior angles.

This rule can be explained by looking at the diagram below:

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From the diagram, we know that angles 'a', 'b' and 'c' all add up to 180° because they are all of the angles within the triangle:

\[a + b + c = {180^ \circ }\]

We also know that angles 'c' and 'd' add up to 180° because they are all the angles on the straight line.

\[c + d = {180^ \circ }\]

By rearranging both equations (subtract c from both sides), you are presented with:

\[a + b = {180^ \circ } - c\]
\[d = {180^ \circ } - c\]
\[a + b = d\]

As a result, you know that the exterior angle 'd' is equal to the sum of the two opposite interior angles 'a' and ‘b’, because they are both equal to 180° - c.

During your GCSE maths exam, you will be required to use these rules to solve various mathematical problems:

Example
(a) - Find the values of x and y in the following triangle:

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Solution
(a) - Using the rule that:

The sum of the two opposite interior angles = the exterior angle

you know that:

\[x + {50^ \circ } = {92^ \circ }\]

From here, you can solve the equation to find x:

\[x = {92^ \circ } - {50^ \circ } = {42^ \circ }\]

Similarly, by using the rule that:

All the angles in a triangle add up to 180°

you know that:

\[y + {42^ \circ } + {50^ \circ } = {180^ \circ }\]

From here, you can solve the equation to find y:

\[y = {180^ \circ } - {50^ \circ } - {42^ \circ } = {180^ \circ } - {92^ \circ } = {88^ \circ }\]

Therefore, x = 42° and y = 88°

2 - Angle properties of quadrilaterals
A quadrilateral is a 4-sided polygon. All of the angles within a quadrilateral add up to 360º. There are several types of quadrilateral; each with different properties:

Square

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  • A square is a regular quadrilateral (i.e. all of its sides are of equal length),
  • All of its angles are equal; 90°,
  • Similarly, all of its sides are of equal length,
  • The opposite sides of a square are parallel,
  • The diagonals of a square bisect one other at 90° and are equal in length,
  • A square has 4 lines of symmetry.

Rhombus

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  • All of the sides of a rhombus are of equal length,
  • Its opposite sides are parallel,
  • Its diagonally opposite angles are equal,
  • Its diagonals bisect one another at 90°,
  • A rhombus has 2 lines of symmetry.

Rectangle

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  • All of the angles in a rectangle are equal; 90°,
  • Its opposite sides are of equal length and are parallel,
  • Its diagonals bisect one another,
  • Its diagonals are equal in length,
  • A rectangle has 2 lines of symmetry.

Parallelogram

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  • A parallelogram has no lines of symmetry,
  • Its diagonally opposite angles are equal,
  • Its opposite sides are of equal length and are parallel,
  • Its diagonals bisect one another.

Trapezium

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  • A trapezium has no lines of symmetry and has no rotational symmetry,
  • One pair of a trapezium's opposite sides is parallel.

Kite

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  • A kite has one line of symmetry and no rotational symmetry,
  • Two pairs of its sides are of equal length,
  • One pair of its diagonally opposite angles is equal,
  • Only one of its diagonals is bisected by the other,
  • A kite's diagonals cross at 90°.

During your GCSE maths exam, you will be required to recognise and identify the properties of these different types of quadrilateral.

3 - Interior angles of polygons
An interior angle is an angle inside a polygon.

The interior angles within different polygons add up to different amounts. For example, as mentioned in the previous sections; the interior angles of a triangle add up to 180° and the interior angles of a quadrilateral add up to 360°.

As a rule, every time a side is added to a polygon, another 180° is added to the interior angle total. The following polygon table demonstrates this rule:

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This rule can be written as:

Sum of Interior Angles = (n-2) × 180°

If a polygon is regular (i.e. all of its sides are of equal length and all of its angles are of equal size), then the following rule is true:

Each Interior Angle of a Regular Polygon = (n-2) × 180° / n

Where 'n' refers to the number of sides.

During your GCSE maths exam, you will be required to use these rules to answer a series of polygon related problems:

Example
(a) - Calculate the sum of the interior angles of a regular octagon and the size of each interior angle

Solution
(a) - An octagon has 8 sides. As a result, the sum of its interior angles will be:

\[(n - 2)*{180^ \circ }\]
\[(8 - 2)*{180^ \circ } = 6*{180^ \circ } = {1080^ \circ }\]

As mentioned in the question, it is a regular octagon, so all of its angles will be of equal size.

Consequently, in order to calculate the size of each interior angle, you can divide the sum of its interior angles by 8:

\[{1080^ \circ }/8 = {135^ \circ }\]

Therefore, the sum of the interior angles in a regular octagon is 1080° and the size of each interior angle is 135°.

4 - Exterior angles of polygons
The exterior angle of a polygon is an angle between any side of a shape and a line extended from the next side.

When you add the exterior angle of a polygon and its corresponding and exterior angle on a straight line, it amounts to 180°.

As a rule;

The sum of the exterior angles of a simple polygon add up to 360°

This is referred to as a 'full revolution'.

The exterior angle of a polygon and its corresponding interior angle always add up to 180°. This is due to the fact that the two corresponding angles create a straight line:

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In order to calculate the exterior angle of a regular polygon, you adhere to the following rule:

Exterior angle of a regular polygon = 360° ÷ (number of sides)

This is due to the fact that the exterior angles of a regular polygon always add up to 360°.

During your GCSE maths exam, you may be required to use these rules to solve various problems:

Example
(a) - The interior angles of a regular polygon are each 120°. Calculate the number of sides.

Solution
(a) - The question tells you that the interior angles of a regular polygon are 120°. As a result, you can calculate the exterior angles. Seeing as an interior angle and its corresponding exterior angle add up to 180°; the size of the exterior angles of this regular polygon:

\[{180^ \circ } - {120^ \circ } = {60^ \circ }\]

By using the rule that the exterior angles of all polygons add up to 360°, you can use this rule to calculate the number of sides:

Exterior angle of a regular polygon = 360° ÷ number of sides

Therefore:

Number of sides = 360° ÷ exterior angle = 360 ° ÷ 60 ° = 6

Therefore, this regular polygon has 6 sides (i.e. it is a hexagon).

Exam Tips

  1. Remember that all of the sides of regular polygon and all of its angles are equal.
  2. Memorise the rule that an interior angle and its corresponding angle add up to 180°, and that all of the exterior angles of a polygon add up to 360°.
  3. Remember that every time a side is added to a polygon, another 180° is added to the interior angle total.
  4. Memorise the rule that the sum of the interior angles of a polygon = (n-2) × 180°, and that each interior angle of a regular polygon can be calculated by the rule: (n-2) × 180° / n, where 'n' refers to the number of sides.

Topic Summary

When revising the topic of 'Polygons', it is highly recommended that you memorise the properties of different types of polygons, as well as their names. For example, as well as memorising the properties of triangles and quadrilaterals, also try to recall the names of larger polygons; such as 7-sided 'heptagons', 8-sided 'octagons', 9-sided 'nonagons' and 10-sided 'decagons'. If you can easily recognise and recall the names and properties of both simple and complex polygons, you will be able to tackle polygon- related exam questions quickly and confidently!

Related Topics

  • Congruent Triangles
  • Trigonometry and Pythagoras
  • Construction and Loci Problems
  • Perimeter, Area, Volume
  • Transformations
  • Circle Theorem Problems
  • Proof