Sectors of circles - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Sectors of circles

Area and volume

Lesson 5 of 7

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Area and volume

Lesson 5 of 7

Answer each one, then check.

1

What is the circumference of a circle with radius 5 cm, in terms of π?

2

What is the area of a circle with radius 6 cm, in terms of π?

3

Simplify 90/360.

4

How many degrees in a full turn?

5

What is the area of a triangle with base 8 and height 8?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Area and volume

Lesson 5 of 7

1

What is the circumference of a circle with radius 5 cm, in terms of π?

10π

2

What is the area of a circle with radius 6 cm, in terms of π?

36π

3

Simplify 90/360.

¼

4

How many degrees in a full turn?

360

5

What is the area of a triangle with base 8 and height 8?

32

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Area and volume

Lesson 5 of 7

1

Identify the arc and sector of a circle.

2

Find arc length and sector area as a fraction of the whole circle.

3

Find the perimeter of a sector.

4

Find the area of a segment (Higher).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Sector and Its Arc

Area and volume

Lesson 5 of 7

A sector is a fraction θ/360 of the whole circle.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Sector Formulae

Area and volume

Lesson 5 of 7

Fraction of the circle

θ/360, where θ is the angle at the centre in degrees.

Arc length

θ/360 × 2πr: a fraction of the circumference.

Sector area

θ/360 × πr²: a fraction of the circle's area.

Sector perimeter

Arc length + 2r (the two straight radii).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Arc Length and Sector Area

Area and volume

Lesson 5 of 7

A sector has radius 6 cm and angle 60°. Find the arc length and the area, in terms of π.

1

Fraction of the circle

60/360 = 1/6

2

Arc length

1/6 × 2π × 6 = 2π

3

Area

1/6 × π × 6² = 6π

ANSWER

Arc length 2π cm (6.28 cm) and area 6π cm² (18.85 cm²).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Perimeter of a Sector

Area and volume

Lesson 5 of 7

A sector has radius 5 cm and angle 90°. Find its perimeter to 1 decimal place.

1

Arc length

90/360 × 2π × 5 = 2.5π= 7.854...

2

Add the two radii

2 × 5 = 10

3

Total

7.854 + 10

ANSWER

17.9 cm

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Finding the Angle

Area and volume

Lesson 5 of 7

A sector has radius 12 cm and an arc length of 8π cm. Find the angle at the centre.

1

Arc length formula

θ/360 × 2π × 12 = 8π

2

Simplify

θ/360 × 24π= 8π

3

Divide both sides by 24π

θ/360 = ⅓

4

Solve

θ= 120

ANSWER

120°

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

HIGHER TIER

The Area of a Segment

Sector minus triangle.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Higher

Segment Area

Area and volume

Lesson 5 of 7

A sector has radius 8 cm and angle 90°. Find the area of the segment cut off by the chord, to 1 decimal place.

1

Sector area

90/360 × π × 8² = 16π= 50.27

2

Triangle area ½r² sin θ

½ × 8 × 8 × sin 90°= 32

3

Subtract

50.27 − 32

ANSWER

18.3 cm²

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Sector or Segment?

Area and volume

Lesson 5 of 7

The words are easy to mix up.

Sector

The "pizza slice" between two radii and an arc.

Segment

The region between a chord and an arc. It is a sector with a triangle cut off.

Segment area

Sector area − triangle area, where the triangle has area ½r²sin θ.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Area and volume

Lesson 5 of 7

Sector

A region of a circle between two radii and the arc joining their ends.

Arc

Part of the circumference of a circle.

Segment

The part of a circle cut off by a chord.

Chord

A straight line joining two points on a circle.

Angle at the centre

The angle between the two radii of a sector.

Fraction of a circle

θ/360.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Area and volume

Lesson 5 of 7

YOUR TASK

Slice the Clock

10 minutes

The minute hand of a clock is 15 cm long. Find the distance its tip travels in 20 minutes, and the area swept out by the hand in 20 minutes. Give both answers in terms of π.

1

Find the fraction of a full turn.

2

Apply it to circumference and area.

WHAT A GOOD ANSWER SHOWS

In 20 minutes the hand turns through 120°, which is ⅓ of a circle. Distance = ⅓ × 30π= 10π cm. Area = ⅓ × 225π= 75π cm².

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Area and volume

Lesson 5 of 7

Write the fraction θ/360.

Find an arc length.

Find a sector area.

Find the perimeter of a sector.

Find the angle from an arc length.

Give answers in terms of π.

Find a segment area (Higher).

Tell a sector from a segment.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Arc length = θ/360 × 2πr.

Sector area = θ/360 × πr².

Sector perimeter = arc + 2r.

Segment area = sector − triangle (Higher).

EXAM FOCUS

A sector of a circle has radius 9 cm and angle 40°. Work out the length of the arc of the sector. Give your answer correct to 3 significant figures. (3 marks)

Write the fraction θ/360 first. Then multiply it by the circumference (arc) or the area (sector).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Sectors of Circles

Area and volume

Lesson 5 of 7

Answer all questions. Show your working. · 30 minutes

Question 1 · 3 marks · Calculator

The diagram shows a sector of a circle of radius 9 cm. The angle at the centre is 40°. Work out the length of the arc. Give your answer…

Question 2 · 3 marks · Calculator

A sector of a circle has radius 10 cm and angle 72°. Work out the area of the sector. Give your answer correct to 3 significant figures.

Question 3 · 4 marks · Calculator

A sector of a circle has radius 5 cm and angle 90°. Work out the perimeter of the sector. Give your answer correct to 3 significant figures.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Sectors of Circles (cont.)

Area and volume

Lesson 5 of 7

Question 4 · 3 marks · Non-calculator

A sector of a circle has radius 12 cm and angle 150°. Work out the length of the arc. Give your answer in terms of π.

Question 5 · 3 marks · Non-calculator

A sector of a circle has radius 12 cm. The arc length of the sector is 8π cm. Work out the angle at the centre of the sector.

Question 6 · 4 marks · Calculator

A sector of a circle has radius 8 cm and angle 90°. Work out the area of the segment formed by the chord joining the ends of the two radii…

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 3 marks · Calculator

Area and volume

Lesson 5 of 7

The diagram shows a sector of a circle of radius 9 cm. The angle at the centre is 40°. Work out the length of the arc. Give your answer correct to 3 significant figures. (3 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Area and volume

Lesson 5 of 7

3 marks available. Award a mark for each point made.

40/360

M1

40/360 × 2 × π × 9

M1

6.28

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 3 marks · Calculator

Area and volume

Lesson 5 of 7

“

A sector of a circle has radius 10 cm and angle 72°. Work out the area of the sector. Give your answer correct to 3 significant figures.

HOW TO ANSWER IT

Command word: Calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · mark scheme

Area and volume

Lesson 5 of 7

3 marks available. Award a mark for each point made.

72/360

M1

72/360 × π × 10²

M1

62.8

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 4 marks · Calculator

Area and volume

Lesson 5 of 7

“

A sector of a circle has radius 5 cm and angle 90°. Work out the perimeter of the sector. Give your answer correct to 3 significant figures.

HOW TO ANSWER IT

Command word: Calculator. Worth 4 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · mark scheme

Area and volume

Lesson 5 of 7

4 marks available. Award a mark for each point made.

Arc length method

M1

7.854

A1

Adding two radii

M1

17.9

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 3 marks · Non-calculator

Area and volume

Lesson 5 of 7

“

A sector of a circle has radius 12 cm and angle 150°. Work out the length of the arc. Give your answer in terms of π.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

Area and volume

Lesson 5 of 7

3 marks available. Award a mark for each point made.

150/360 or 5/12

M1

5/12 × 24π

M1

10π

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 3 marks · Non-calculator

Area and volume

Lesson 5 of 7

“

A sector of a circle has radius 12 cm. The arc length of the sector is 8π cm. Work out the angle at the centre of the sector.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · mark scheme

Area and volume

Lesson 5 of 7

3 marks available. Award a mark for each point made.

θ/360 × 2π × 12 = 8π

M1

θ/360 = ⅓

M1

120

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 4 marks · Calculator

Area and volume

Lesson 5 of 7

“

A sector of a circle has radius 8 cm and angle 90°. Work out the area of the segment formed by the chord joining the ends of the two radii. Give your answer correct to 3 significant figures.

HOW TO ANSWER IT

Command word: Calculator. Worth 4 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · mark scheme

Area and volume

Lesson 5 of 7

4 marks available. Award a mark for each point made.

Sector area 50.27

M1

Triangle area 32

M1

50.27 − 32

M1

18.3

A1

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