Pyramids and cones - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Pyramids and cones

Area and volume

Lesson 7 of 7

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Area and volume

Lesson 7 of 7

Answer each one, then check.

1

What is the area of a square with side 6?

2

State Pythagoras' theorem.

3

What is ⅓ of 36 multiplied by 9?

4

What is the area of a circle with radius 4, in terms of π?

5

What is the volume of a prism?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Area and volume

Lesson 7 of 7

1

What is the area of a square with side 6?

36

2

State Pythagoras' theorem.

a² + b² = c²

3

What is ⅓ of 36 multiplied by 9?

108

4

What is the area of a circle with radius 4, in terms of π?

16π

5

What is the volume of a prism?

Cross-sectional area times length

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Area and volume

Lesson 7 of 7

1

Find the volume of a pyramid using ⅓ × base area × h.

2

Find the volume and curved surface area of a cone.

3

Use Pythagoras to find a slant height.

4

Find the volume of a frustum (Higher).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Pyramids and Cones

Area and volume

Lesson 7 of 7

The perpendicular height is not the slant height.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Area and volume

Lesson 7 of 7

THE KEY FACT

A pyramid or cone has one third of the volume of the prism or cylinder with the same base and height.

V = ⅓ × base area × perpendicular height

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Pyramid

Area and volume

Lesson 7 of 7

A square-based pyramid has base side 6 cm and perpendicular height 9 cm. Find its volume.

1

Area of the base

6 × 6 = 36

2

Volume = ⅓ × base area × h

⅓ × 36 × 9

3

Work out

108

ANSWER

108 cm³

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Cone

Area and volume

Lesson 7 of 7

A cone has base radius 4 cm and perpendicular height 9 cm. Find its volume in terms of π and to 1 decimal place.

1

Volume = ⅓πr² h

⅓ × π × 16 × 9

2

Simplify

48π

3

As a decimal

150.8

ANSWER

48π= 150.8 cm³

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Slant Height and Curved Surface

Area and volume

Lesson 7 of 7

A cone has base radius 5 cm and perpendicular height 12 cm. Find (a) the slant height, and (b) the curved surface area in terms of π.

1

(a) The radius, height and slant height form a right-angled triangle

l² = 5² + 12² = 169

2

Slant height

l = 13

3

(b) Curved surface area = πr l

π × 5 × 13 = 65π

ANSWER

(a) 13 cm (b) 65π cm² (204.2 cm²)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Formulae for Pyramids and Cones

Area and volume

Lesson 7 of 7

The volume formula for a cone is on the formulae sheet; the curved surface area of a cone is too.

Solid

Volume

Surface

Pyramid

⅓ × base area × h

Add the areas of all the faces

Cone

⅓πr² h

Curved surface πr l; base πr²

Slant height

l = √(r² + h²)

Pythagoras in the cone

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

HIGHER TIER

Frustums

A cone with its top cut off.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Higher

Volume of a Frustum

Area and volume

Lesson 7 of 7

A cone of radius 6 cm and height 12 cm has a small cone of radius 3 cm and height 6 cm cut off the top. Find the volume of the frustum that is left, in terms of π.

1

Volume of the whole cone

⅓ × π × 6² × 12 = 144π

2

Volume of the small cone

⅓ × π × 3² × 6 = 18π

3

Subtract

144π− 18π

ANSWER

126π cm³ (395.8 cm³)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Area and volume

Lesson 7 of 7

Pyramid

A solid with a polygon base and triangular faces meeting at a point.

Apex

The point at the top of a pyramid or cone.

Cone

A solid with a circular base and a curved surface up to an apex.

Perpendicular height

The height measured at right angles to the base.

Slant height

The distance from the apex down the side of a cone to the base edge.

Frustum

What is left when the top of a cone or pyramid is cut off parallel to the base.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Area and volume

Lesson 7 of 7

YOUR TASK

Ice Cream Cone

12 minutes

A cone has radius 3 cm and slant height 10 cm. Work out the perpendicular height, the volume of the cone, and the volume of a hemisphere of ice cream of radius 3 cm on top. Will the ice cream fit inside the cone if it melts?

1

Use Pythagoras for the height.

2

Use each volume formula.

3

Compare.

WHAT A GOOD ANSWER SHOWS

Height = √(100 − 9) = 9.54 cm. Cone volume = ⅓π × 9 × 9.54 = 89.9 cm³. Hemisphere = ⅔π × 27 = 56.5 cm³. The ice cream volume is less than the cone volume, so it would fit.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Area and volume

Lesson 7 of 7

Find the volume of a pyramid.

Find the volume of a cone.

Use Pythagoras to find a slant height.

Find the curved surface area of a cone.

Find the total surface area of a cone.

Work backwards to find a height.

Find the volume of a frustum (Higher).

Give answers in terms of π.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Pyramid and cone volume: one third of base area times perpendicular height.

Cone curved surface πr l, with l = √(r² + h²).

Do not confuse perpendicular height with slant height.

Frustum: whole cone minus small cone (Higher).

EXAM FOCUS

A cone has radius 5 cm and vertical height 12 cm. Work out the curved surface area of the cone. Give your answer in terms of π. (3 marks)

Find the slant height with Pythagoras before you use πr l. The vertical height goes in the volume formula; the slant height goes in the surface area formula.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Pyramids and Cones

Area and volume

Lesson 7 of 7

Answer all questions. Show your working. · 30 minutes

Question 1 · 2 marks · Non-calculator

A square-based pyramid has a base of side 6 cm and a perpendicular height of 9 cm. Work out the volume of the pyramid.

Question 2 · 2 marks · Calculator

A cone has a base radius of 4 cm and a perpendicular height of 9 cm. Work out the volume of the cone. Give your answer correct to 3…

Question 3 · 3 marks · Non-calculator

The diagram shows a cone with base radius 5 cm and vertical height 12 cm. Work out the curved surface area of the cone. Give your answer in…

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Pyramids and Cones (cont.)

Area and volume

Lesson 7 of 7

Question 4 · 2 marks · Calculator

Work out the volume of the cone in the last question. Give your answer correct to 3 significant figures.

Question 5 · 4 marks · Calculator

A cone has base radius 6 cm and height 12 cm. A smaller cone of base radius 3 cm and height 6 cm is cut off the top, leaving a frustum…

Question 6 · 2 marks · Non-calculator

A pyramid has a rectangular base measuring 8 cm by 6 cm and a perpendicular height of 5 cm. Work out the volume of the pyramid.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 2 marks · Non-calculator

Area and volume

Lesson 7 of 7

“

A square-based pyramid has a base of side 6 cm and a perpendicular height of 9 cm. Work out the volume of the pyramid.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Area and volume

Lesson 7 of 7

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

⅓ × 6 × 6 × 9

M1

108

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 2 marks · Calculator

Area and volume

Lesson 7 of 7

“

A cone has a base radius of 4 cm and a perpendicular height of 9 cm. Work out the volume of the cone. Give your answer correct to 3 significant figures.

HOW TO ANSWER IT

Command word: Calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · mark scheme

Area and volume

Lesson 7 of 7

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

⅓π × 4² × 9

M1

151

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 3 marks · Non-calculator

Area and volume

Lesson 7 of 7

The diagram shows a cone with base radius 5 cm and vertical height 12 cm. Work out the curved surface area of the cone. Give your answer in terms of π. (3 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · mark scheme

Area and volume

Lesson 7 of 7

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

5² + 12²

M1

Slant height 13

A1

65π

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 2 marks · Calculator

Area and volume

Lesson 7 of 7

“

Work out the volume of the cone in the last question. Give your answer correct to 3 significant figures.

HOW TO ANSWER IT

Command word: Calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

Area and volume

Lesson 7 of 7

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

⅓π × 5² × 12

M1

314

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 4 marks · Calculator

Area and volume

Lesson 7 of 7

“

A cone has base radius 6 cm and height 12 cm. A smaller cone of base radius 3 cm and height 6 cm is cut off the top, leaving a frustum. Work out the volume of the frustum. 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 5 · mark scheme

Area and volume

Lesson 7 of 7

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

Volume of the large cone

M1

Volume of the small cone

M1

Subtracting

M1

396

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 2 marks · Non-calculator

Area and volume

Lesson 7 of 7

“

A pyramid has a rectangular base measuring 8 cm by 6 cm and a perpendicular height of 5 cm. Work out the volume of the pyramid.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · mark scheme

Area and volume

Lesson 7 of 7

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

⅓ × 8 × 6 × 5

M1

80

A1

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