Exterior angles of a polygon.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exterior angles of a polygon

Angles and trigonometry

Lesson 3 of 7

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before

Angles and trigonometry

Lesson 3 of 7

Answer each one, then check.

1

Last lesson: what do the interior angles of a pentagon add up to?

2

What do angles on a straight line add up to?

3

Work out 360 ÷ 8.

4

What is each interior angle of a regular hexagon?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before - Answers

Angles and trigonometry

Lesson 3 of 7

1

Last lesson: what do the interior angles of a pentagon add up to?

540°

2

What do angles on a straight line add up to?

180°

3

Work out 360 ÷ 8.

45

4

What is each interior angle of a regular hexagon?

120°

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Angles and trigonometry

Lesson 3 of 7

1

Know that the exterior angles of any polygon add up to 360°.

2

Find the exterior angle of a regular polygon.

3

Use interior angle + exterior angle = 180°.

4

Find the number of sides of a regular polygon from one of its angles.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Full Turn

Angles and trigonometry

Lesson 3 of 7

The exterior angles fit together to make one full turn.

An exterior angle is the angle between one side and the extension of the next side. Walk round the shape and you turn by each exterior angle in turn - and arrive facing the way you started. That is one full turn, so the exterior angles add up to 360° for every polygon.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exterior Angle Facts

Angles and trigonometry

Lesson 3 of 7

Two facts do almost all the work.

The sum

The exterior angles of any polygon add up to 360°.

Interior and exterior

At each vertex, interior angle + exterior angle = 180° (angles on a straight line).

Regular polygon

Each exterior angle = 360/n.

Number of sides

n = 360/(exterior angle).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Angles of a Regular Pentagon

Angles and trigonometry

Lesson 3 of 7

Work out the exterior and interior angles of a regular pentagon.

1

Exterior angle

360 ÷ 5 = 72°

2

Interior angle: they add up to 180°

180 − 72 = 108°

3

Check with last lesson's formula

540 ÷ 5 = 108°

ANSWER

Exterior 72°, interior 108°

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

How Many Sides?

Angles and trigonometry

Lesson 3 of 7

Each interior angle of a regular polygon is 160°. How many sides does it have?

1

Find the exterior angle

180 − 160 = 20°

2

Exterior angles add up to 360°

360 ÷ 20 = 18

ANSWER

18 sides

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Which Method?

Angles and trigonometry

Lesson 3 of 7

Exterior angles are usually quicker for regular polygons.

1

Regular, angles wanted

Exterior = 360 ÷ n, then interior = 180 − exterior.

2

Regular, sides wanted

Exterior = 180 − interior, then n = 360 ÷ exterior.

3

Irregular

Use the angle sum (n − 2) × 180, or exterior angles adding up to 360°.

4

Shapes joined together

Use angles around a point: they add up to 360°.

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Polygons Meeting at a Point

Angles and trigonometry

Lesson 3 of 7

A regular hexagon and a square share a side, and meet at a point. Work out the angle x in the gap between them at that point.

1

Interior angle of a regular hexagon

180 − 360 ÷ 6 = 120°

2

Interior angle of a square

90°

3

Angles around a point add up to 360°

x = 360 − 120 − 90 = 150

ANSWER

x = 150°

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Angles and trigonometry

Lesson 3 of 7

Exterior angle

The angle between one side of a polygon and the extension of the side next to it.

Interior angle

The angle inside a polygon at a vertex.

Angles around a point

Angles meeting at a point; they add up to 360°.

Regular polygon

A polygon with all sides and all angles equal.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Angles and trigonometry

Lesson 3 of 7

YOUR TASK

Impossible Polygons

10 minutes

For each angle, decide whether it could be the exterior angle of a regular polygon. If it could, say how many sides the polygon has: 40°, 50°, 24°, 70°, 15°, 1°.

1

Divide 360 by the angle.

2

A whole number means it is possible.

3

Write down the number of sides.

WHAT A GOOD ANSWER SHOWS

40°: 9 sides. 50°: no (7.2). 24°: 15 sides. 70°: no. 15°: 24 sides. 1°: 360 sides. An exterior angle works only if it divides exactly into 360.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Angles and trigonometry

Lesson 3 of 7

Recall that exterior angles add up to 360°.

Find the exterior angle of a regular polygon.

Use interior + exterior = 180°.

Find the number of sides from an angle.

Decide whether an angle is possible for a regular polygon.

Solve problems with polygons meeting at a point.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Exterior angles of any polygon add up to 360°.

Regular polygon: exterior angle = 360 ÷ n, and n = 360 ÷ exterior angle.

Interior + exterior = 180°.

EXAM FOCUS

The diagram shows a regular hexagon and a square that share a side. Work out the size of angle x. (3 marks)

For any regular polygon question, work out the exterior angle first: 360 ÷ n. Everything else follows from it.

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