Interior angles of a polygon.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Interior angles of a polygon

Angles and trigonometry

Lesson 2 of 7

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before

Angles and trigonometry

Lesson 2 of 7

Answer each one, then check.

1

Last lesson: what do the angles in a quadrilateral add up to?

2

How many sides does a hexagon have?

3

Work out 8 × 180.

4

Solve x/5 = 108.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before - Answers

Angles and trigonometry

Lesson 2 of 7

1

Last lesson: what do the angles in a quadrilateral add up to?

360°

2

How many sides does a hexagon have?

6

3

Work out 8 × 180.

1440

4

Solve x/5 = 108.

x = 540

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Angles and trigonometry

Lesson 2 of 7

1

Name polygons up to 10 sides.

2

Work out the sum of the interior angles of any polygon.

3

Find a missing interior angle.

4

Work out the interior angle of a regular polygon.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Polygons

Angles and trigonometry

Lesson 2 of 7

A polygon is a 2D shape with straight sides.

Names

Triangle 3, quadrilateral 4, pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10.

Interior angle

An angle inside the polygon, at a vertex (corner).

Regular

All sides equal AND all angles equal.

Irregular

Sides or angles not all equal - the angle sum is still the same.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Why the Formula Works

Angles and trigonometry

Lesson 2 of 7

An n-sided polygon splits into n − 2 triangles.

Draw lines from one vertex to every other vertex. A polygon with n sides always splits into n − 2 triangles, and each triangle holds 180°. So the interior angles add up to (n − 2) × 180°.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Interior Angle Sums

Angles and trigonometry

Lesson 2 of 7

Polygon

Sides

Triangles

Angle sum

Triangle

3

1

180°

Quadrilateral

4

2

360°

Pentagon

5

3

540°

Hexagon

6

4

720°

Octagon

8

6

1080°

Decagon

10

8

1440°

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Missing Angle in a Pentagon

Angles and trigonometry

Lesson 2 of 7

Four of the angles of a pentagon are 100°, 120°, 95° and 110°. Work out the fifth angle.

1

Angle sum of a pentagon

(5 − 2) × 180 = 540

2

Add the known angles

100 + 120 + 95 + 110 = 425

3

Subtract

540 − 425 = 115

ANSWER

115°

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Regular Polygons

Angles and trigonometry

Lesson 2 of 7

In a regular polygon every interior angle is the same, so share the angle sum equally.

Interior angle

Interior angle of a regular polygon = ((n − 2) × 180)/n.

Regular hexagon

720/6 = 120°.

Regular octagon

1080/8 = 135°.

Growing

The more sides, the closer the interior angle gets to 180° - but it never reaches it.

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The Interior Angle of a Regular Decagon

Angles and trigonometry

Lesson 2 of 7

Work out the size of each interior angle of a regular decagon.

1

A decagon has 10 sides

n = 10

2

Angle sum

(10 − 2) × 180 = 1440

3

Share equally between 10 angles

1440 ÷ 10 = 144

ANSWER

144°

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Working Back to the Number of Sides

Angles and trigonometry

Lesson 2 of 7

Each interior angle of a regular polygon is 150°. How many sides does it have?

1

Write the angle sum two ways

(n − 2) × 180 = 150n

2

Expand

180n − 360 = 150n

3

Solve

30n = 360, so n = 12

ANSWER

12 sides (a dodecagon). Next lesson shows a quicker way.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Angles and trigonometry

Lesson 2 of 7

Polygon

A closed 2D shape with straight sides.

Vertex

A corner of a shape (plural: vertices).

Interior angle

An angle inside a polygon at a vertex.

Regular polygon

A polygon with all sides and all angles equal.

Diagonal

A straight line joining two vertices that are not next to each other.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Angles and trigonometry

Lesson 2 of 7

YOUR TASK

Does It Tessellate?

10 minutes

A regular polygon tessellates (tiles a flat surface with no gaps) if copies of it fit exactly around a point. Work out the interior angles of the regular triangle, square, pentagon, hexagon and octagon, and decide which of them tessellate on their own.

1

Work out each interior angle.

2

Check whether it divides exactly into 360°.

3

Explain your answer.

WHAT A GOOD ANSWER SHOWS

Interior angles: 60, 90, 108, 120, 135. Only the triangle (6 × 60 = 360), square (4 × 90) and hexagon (3 × 120) divide exactly into 360°, so only those three tessellate.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Angles and trigonometry

Lesson 2 of 7

Name polygons up to 10 sides.

Explain why the angle sum is (n − 2) × 180°.

Find the angle sum of any polygon.

Find a missing interior angle.

Find the interior angle of a regular polygon.

Find the number of sides from the interior angle.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Angle sum = (n − 2) × 180°.

Regular polygon: each interior angle = ((n − 2) × 180)/n.

Missing angle: angle sum minus the angles you know.

EXAM FOCUS

Work out the size of each interior angle of a regular nonagon (9 sides). (2 marks)

Write the formula before you substitute. If you only remember one thing, remember that a polygon splits into n − 2 triangles - you can rebuild the formula from that.

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