GCSE Maths Geometry and Measure

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.

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 and angles that are not equal to each other.
• The table below gives the name of several irregular polygon.
• The sum of interior angles can be calculated using the formula:

Sum of interior angles = (n-2) × 180

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.

## How to classify an irregular polygon

In order to classify an irregular polygon:

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

### How to classify an irregular polygon ## Irregular polygons examples

### Example 1: irregular triangles

Given that ABC is an isosceles triangle, classify the polygon ABD .

1. 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) 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.

Classify the following polygon:

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

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

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

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.

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

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 .

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.

The pentagon has 5 sides.

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 × 2 = 106° .

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.

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

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.

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.

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

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=412+16=64°

6x+8=612+8=80°

2x+12=212+12=36°

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°

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

1. BD is parallel to AE in the triangle ACE . Classify the polygon BCD 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.

2. A quadrilateral is inscribed in a circle with centre O . The diagonals meet perpendicularly at the point E where AC = 2AE . Use circle theorems to determine the classification of the quadrilateral ABCD . Diamond  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.

3. The polygon below has an order of rotational symmetry of 4 . Classify the polygon. Octagon  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.

4. 6 congruent triangles are joined together to form a 6 sided polygon. Each line segment within the polygon is the same length. The new polygon has two acute angles, 3 obtuse angles and one reflex angle. The polygon has one line of symmetry and no rotational symmetry. Classify the polygon. Regular hexagon Hexagon Parallelogram As the polygon has 6 sides and different interior angles, the polygon is a hexagon.

5. The polygon below is incomplete. Given that the polygon is made up of 10 congruent kites, classify the polygon. 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.

6. The interior angles of a polygon are: 12x-30, 8x+40, 7x+30, 8x, and 5x+20 . The sum of the angles is equal to 540^{\circ} . Classify the polygon.

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=712+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)

(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)

(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) The angle OBC = 60^{\circ} . Calculate the size of angle OAB .

(5 marks)

(a)
8x^{2}

(1)

(b)
x^{2}=40

(1)

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

(1)

(c)
ABO = 603=20^{\circ}

(1)

BAO = 180 – (90 +20) = 70^{\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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