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Angles in polygons Types of quadrilaterals Parallel lines

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GCSE Maths Geometry and Measure 2D Shapes Polygons

Irregular polygon

Irregular Polygons

Here we will learn about polygons, including regular polygons, angles in polygons, and complex polygons.

There are also polygons worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is an irregular polygon?

An irregular polygon is a 2D shape that has straight sides that are not equal to each other and angles that are not equal to each other. You need to be able to classify geometric shapes based on their properties and sizes and find unknown angles in any polygon. To do this, we need to look closely at the properties of these shapes.

E.g.


Trapezium	Irregular Pentagon	Irregular dodecagon

To do this, we need to look closely at the properties of these shapes.

What is an irregular polygon?

Properties of irregular polygons

Irregular polygons have sides that are not equal to each other or angles that are not equal to each other or both.
The table below gives the name of several irregular polygon.

Polygon	Number of sides	Image (irregular polygon)
Scalene triangle	3
Irregular quadrilateral	4
Irregular pentagon	5
Irregular hexagon	6
Irregular heptagon	7
Irregular octagon	8
Irregular nonagon	9
Irregular decagon	10
Irregular hendecagon	11
Irregular dodecagon	12

The sum of interior angles can be calculated using the formula:

Sum of interior angles = $(n-2) × 180^{\circ}$

where $n$ represents the number of sides.

As the size of each angle is not equal, we can determine the size of each angle by adding together the interior angles that we know and putting it equal to the sum of the interior angles, then solving the equation.

Exterior angles are supplementary to the interior angle.

Sum of exterior angles of a polygon = $360°$

We can use this property to find either the interior angle, or exterior angle at a vertex.

As the size of each angle is not equal, we can determine the size of an exterior angle by adding together the exterior angles that we know and subtracting it from $360°$ .

Irregular polygons can be either convex polygons or concave polygons.

A convex polygon is a polygon with all the interior angles less than $180°$ . A concave polygon has at least one angle that is greater than $180°$ (or a reflex angle).

E.g.

Convex hexagon	Concave hexagon
All interior angles are either acute or obtuse.	At least one angle is greater than 180°

How to classify an irregular polygon

In order to classify an irregular polygon:

State/calculate the number of sides of the polygon.
Determine the size of the angles and side lengths within the polygon.
Recognise the other properties of the polygon.

How to classify an irregular polygon

Irregular polygons worksheet

Get your free Irregular polygons worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Irregular polygons worksheet

Get your free Irregular polygons worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Related lessons on polygons

Irregular polygon is part of our series of lessons to support revision on polygons. You may find it helpful to start with the main polygons lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Irregular polygons examples

Example 1: irregular triangles

Given that $ABC$ is an isosceles triangle and $BCD$ is a straight line, classify the polygon $ABD$ .

State/calculate the number of sides of the polygon.

$ABC$ has $3$ sides and so it is a triangle.

2Determine the size of the angles and/or side lengths within the polygon.

As $BCA = 116°$ , angle $ACD = 180 - 116 = 64°$ .

As $ABC$ is an isosceles triangle, $ABC = BAC = (180 - 116)\div 2 = 32°$ .

As angles in a triangle total $180°$ , angle $CAD = 180 - (74 + 64) = 42°$ .

Angle $BAD = 42 + 32 = 74°$ .

3Recognise the other properties of the polygon.

As the three angles in the triangle $ABC$ are $74°, 74°$ and $32°$ . The triangle $ABD$ has two equal angles and so it is isosceles.

The polygon is an isosceles triangle, which is an irregular polygon.

Example 2: irregular quadrilateral

Classify the following polygon:

State/calculate the number of sides of the polygon.

The polygon has $4$ sides so this is a type of quadrilateral.

Determine the size of the angles and/or side lengths within the polygon.

Angle $ACD$ is alternate to angle $BAC$ and so angle $ACD = 48°$ .

The angle $COD = 180 - (41 + 48) = 91°$ .

Recognise the other properties of the polygon.

As the angle at the centre is not equal to $90$ degrees and the four sides are two pairs of parallel lines, the polygon $ABCD$ is a parallelogram, which is an irregular quadrilateral.

Example 3: types of trapezium

The polygon below is a quadrilateral. By using angle properties, determine the type of quadrilateral.

State/calculate the number of sides of the polygon.

It is stated in the question that we have a quadrilateral.

Determine the size of the angles and/or side lengths within the polygon.

We do not have any angles to use to classify this polygon. This in turn discounts many quadrilaterals that have specific angle properties. We can however use the information provided to find the length of $AD$ .

As the vertical height is perpendicular to the base, we can use Pythagoras’ Theorem to calculate the length of $AD$ .

$3^{2}+4^{2}=AD^{2}$

$9+16=AD^{2}$

$AD^{2}=25$

$AD=5cm$

Recognise the other properties of the polygon.

As $AD$ does not equal $BC$ but $AB$ and $CD$ are parallel, the quadrilateral is a trapezium (which is an irregular polygon).

Example 4: pentagon

A pentagon is constructed by joining an isosceles triangle with a rectangle. The line $AM$ is a line of symmetry. Show that the polygon is irregular.

State/calculate the number of sides of the polygon.

The pentagon has $5$ sides.

Determine the size of the angles and/or side lengths within the polygon.

As angles in a triangle total $180°$ , angle $AEB = 180 - (90+53) = 37°$ .

As $BCDE$ is a rectangle, angle $BED = 90°$ and so angle $AED = 90 + 37 = 127°$ .

As the polygon is symmetrical, angle $BAE = 53 \times 2 = 106°$ .

Recognise the other properties of the polygon.

The interior angles of the polygon are equal to $106°, 120°, 90°, 106°$ and $120°$ so, as the angles are not the same, the pentagon is irregular.

Example 5: concave polygon

Below is a description of a polygon. From the description, determine the classification of the polygon.

The polygon has $10$ edges.

The polygon has a rotational symmetry of order $5$ .

The interior angles are equal to $42°$ or $246°$ .

The polygon can be constructed from $5$ congruent kites.

State/calculate the number of sides of the polygon.

As the polygon has $10$ edges, this polygon is a type of decagon.

Determine the size of the angles and/or side lengths within the polygon.

As the polygon can be constructed from $5$ congruent kites, each side length of the decagon is the same. As the polygon has a rotational symmetry of order $5$ and the interior angles are $42°$ or $246°$ , the polygon looks like a five pointed star.

Recognise the other properties of the polygon.

As the angles in the polygon are not equal, the shape is an irregular decagon.

Example 6: algebraic

The interior angles of a polygon are: $4x+16, 6x+8$ and $2x+12$ . Classify the polygon.

State/calculate the number of sides of the polygon.

As there are $3$ interior angles, the polygon is a type of triangle.

Determine the size of the angles and/or side lengths within the polygon.

As the sum of angles in a triangle total $180°$

$4x+16+6x+8+2x+12=180$

$12x+36=180$

$12x=144$

$x=12$

As $x=12$

$4x+15=4\times 12+16=64°$

$6x+8=6\times 12+8=80°$

$2x+12=2\times 12+12=36°$

Recognise the other properties of the polygon.

As all the angels in the polygon are different, the triangle is a scalene triangle (an irregular polygon).

Common misconceptions

Angles in polygons

Make sure you know your angle properties. Getting these confused causes quite a few misconceptions.

Angles in a triangle total $180°$
Angles in a quadrilateral total $360°$
Angles on a straight line total $180°$

Incorrect quadrilateral classification

There are many quadrilaterals and it is common to confuse the properties, especially for a rhombus, parallelogram, or trapezium.

As well as this, stating that the polygon is a quadrilateral is not enough information for a classification.

Incorrect assumptions for triangles

Assuming a triangle is isosceles or equilateral can have an impact on the size of angles within the rest of the polygon, so make sure you can explain why you have chosen a specific type of triangle.

As well as this, stating that the polygon is a triangle is not enough information for a classification. You must state whether it is isosceles/ equilateral etc.

Practice Irregular polygons questions

Equilateral triangle

Right Angle triangle

Isosceles triangle

Scalene triangle

The angle $BDC = 180 – 127 = 53^{\circ}$ (angles on a straight line total $180^{\circ}$ ).
Angle $CBD = 180 – (65 + 53) = 62^{\circ}$ (angles in a triangle total $180^{\circ}$ ).
All angles of $BCD$ are different.

Diamond

Arrowhead

Isosceles triangle

Kite

As $AC = 2AE$ , and $BD$ is perpendicular to $AC$ , triangles $ABC$ and $ADC$ are isosceles triangles. This means that the two angles at $A$ and $C$ are equal and the side lengths $AB$ and $BC$ , and $AD$ and $CD$ are equal. This is a kite.

Octagon

Quadrilateral

Regular octagon

Regular pentagon

As the order of rotational symmetry is $4$ , the angles at $A, C, E$ and $G$ are all equal to $40^{\circ}$ , and the angles at $B, D, F,$ and $H$ are equal to $360 – 130 = 230^{\circ}$ . The polygon has $8$ sides but the interior angles are not equal.

Arrowhead

Regular hexagon

Hexagon

Parallelogram

As the polygon has $6$ sides and different interior angles, the polygon is a hexagon.

Regular decagon

Regular dodecagon

Regular icosagon

Regular octagon

As the polygon is made of $10$ congruent kites, each side length is the same and the interior angles are also equal due to the kite also being symmetrical.

Pentagon

Hexagon

Regular pentagon

Regular hexagon

12x-30+8x+40+7x+30+8x+5x+20=540\\ 40x+60=540\\ 40x=480\\ x=12

As $x=12\\ 12x-30=12\times12-30=114^{\circ}\\ 8x+40=8\times12+40=136^{\circ}\\ 7x+30=7\times12+30=114^{\circ}\\ 8x=8\times12=96^{\circ}\\ 5x+20=5\times12+20=80^{\circ}$

The internal angles are not equal and so the polygon is a pentagon.

Irregular polygons GCSE questions

1. (a) An isosceles trapezium is made when you join two triangles together along an edge with the same length. $ABC$ is an isosceles triangle. Work out the size of the angle $ACD$ .

(b) What type of triangle is $ACD$ ?

(5 marks)

Show answer

(a)
$BCA = (180 – 116)\div2 = 32^{\circ}$

(1)

CAD = 116 – 32 = 84^{\circ}

(1)

ADC = 180 – 116 = 64^{\circ}

(1)

ACD = 180 – (84+64) = 32^{\circ}

(1)

(b)
Scalene

(1)

2. (a) Show that the triangle is a right angle triangle.

(b)The area of both polygons are equal. Calculate the length $x$ .

(5 marks)

Show answer

(a)
$5^{2}+12^{2}=25+144=169$

(1)

\sqrt{169}=13

(1)

The sum of the square of the two shorter sides is equal to the square of the hypotenuse so the triangle contains a right angle

(1)

(b)
Triangle area = $5\times12\div2=30cm^{2}$

(1)

30\div6=x\\ x=5cm

(1)

3. (a) The polygon $ABC$ is half of a kite where $AC$ is the longest length of the kite. Write an expression for the area of the kite.

(b) Given that the area of the smaller triangle $ABO = 40cm^{2}$ , calculate the area of the kite.
(c) Calculate the size of angle $OAB$ .

(5 marks)

Show answer

(a)
$8x^{2}$

(1)

(b)
$x^{2}=40$

(1)

8x^{2}=320cm^{2}

(1)

OAB = 63.4^{\circ}

(1)

Learning checklist

You have now learned how to:

Plot specified points and draw sides to complete a given polygon

Distinguish between regular and irregular polygons based on reasoning about equal sides and angles

Compare and classify geometric shapes based on their properties and sizes and find unknown angles in any triangles, quadrilaterals, and regular polygons

Describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively and rotationally symmetric

Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons

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