Cylinders and spheres - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Cylinders and spheres

Area and volume

Lesson 6 of 7

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Area and volume

Lesson 6 of 7

Answer each one, then check.

1

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

2

What is the circumference of a circle with diameter 10, in terms of π?

3

Work out 5³.

4

What is the volume of a prism?

5

Round 3.979 to 2 decimal places.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Area and volume

Lesson 6 of 7

1

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

16π

2

What is the circumference of a circle with diameter 10, in terms of π?

10π

3

Work out 5³.

125

4

What is the volume of a prism?

Cross-sectional area times length

5

Round 3.979 to 2 decimal places.

3.98

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Area and volume

Lesson 6 of 7

1

Find the volume and surface area of a cylinder.

2

Find the volume and surface area of a sphere.

3

Find the volume and surface area of a hemisphere.

4

Solve problems that combine cylinders and hemispheres.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Cylinder, Sphere and Hemisphere

Area and volume

Lesson 6 of 7

A cylinder is a prism with a circular cross-section.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Formulae

Area and volume

Lesson 6 of 7

The sphere formulae are given on the Edexcel formulae sheet, but you must know the cylinder ones.

Solid

Volume

Surface area

Cylinder

πr² h

2πr² + 2πr h

Sphere

4/3πr³

4πr²

Hemisphere

⅔πr³

3πr²

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Cylinder

Area and volume

Lesson 6 of 7

A cylinder has radius 5 cm and height 12 cm. Find its volume and total surface area, in terms of π and to 1 decimal place.

1

Volume = πr² h

π × 25 × 12 = 300π

2

As a decimal

942.5 cm³

3

Curved surface = 2πr h

2π × 5 × 12 = 120π

4

Two circular ends = 2πr²

2π × 25 = 50π

5

Total surface area

170π= 534.1 cm²

ANSWER

Volume 300π cm³ (942.5) and surface area 170π cm² (534.1).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Sphere

Area and volume

Lesson 6 of 7

A sphere has radius 6 cm. Find its volume and surface area, in terms of π.

1

Volume = 4/3πr³

4/3 × π × 216 = 288π

2

Surface area = 4πr²

4 × π × 36 = 144π

ANSWER

Volume 288π cm³ and surface area 144π cm².

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Hemisphere

Area and volume

Lesson 6 of 7

A solid hemisphere has radius 3 cm. Find its volume in terms of π.

1

A hemisphere is half a sphere

½ × 4/3πr³ = ⅔πr³

2

Substitute r = 3

⅔ × π × 27

3

Work out

18π

ANSWER

18π cm³

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Finding the Height

Area and volume

Lesson 6 of 7

A cylinder has volume 200 cm³ and radius 4 cm. Find its height to 2 decimal places.

1

πr² h = 200

π × 16 × h = 200

2

Divide by 16π

h = 200/16π

3

Calculate

3.979...

ANSWER

3.98 cm

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Combined Solids

Area and volume

Lesson 6 of 7

Add the volumes; for surface area, do not count the joined faces.

A capsule

A cylinder with a hemisphere at each end: total volume = πr² h + 4/3πr³.

A hemisphere on top

Surface area is the cylinder's curved surface and one end, plus the hemisphere's curved surface 2πr².

Watch the radius

The radius of the hemisphere is the radius of the cylinder.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Area and volume

Lesson 6 of 7

Cylinder

A prism with a circle as its cross-section.

Sphere

A perfectly round 3D shape, like a ball.

Hemisphere

Half a sphere.

Curved surface area

The area of the curved part only, not the flat ends.

Volume

The amount of space inside a solid.

In terms of π

Leaving π in the answer.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Area and volume

Lesson 6 of 7

YOUR TASK

Tennis Ball Tube

12 minutes

Three tennis balls of radius 3.3 cm fit exactly in a cylindrical tube. Find the volume of the tube, the volume of the three balls, and the percentage of the tube filled with balls.

1

Find the height of the tube.

2

Work out both volumes.

3

Divide and convert to a percentage.

WHAT A GOOD ANSWER SHOWS

The tube has radius 3.3 and height 19.8: π × 3.3² × 19.8 = 677.4 cm³. The balls: 3 × 4/3π × 3.3³ = 4π × 35.937 = 451.6 cm³. Percentage = 451.6 ÷ 677.4 = 66.7%, exactly two thirds.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Area and volume

Lesson 6 of 7

Find the volume of a cylinder.

Find the surface area of a cylinder.

Find the volume of a sphere.

Find the surface area of a sphere.

Find the volume of a hemisphere.

Work backwards to find a radius or height.

Solve problems with combined solids.

Give answers in terms of π.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Cylinder volume πr² h; curved surface 2πr h.

Sphere volume 4/3πr³; surface 4πr².

Hemisphere volume ⅔πr³.

For combined solids, add volumes and take care with joined faces.

EXAM FOCUS

A cylinder has radius 5 cm and height 12 cm. Work out the volume of the cylinder. Give your answer correct to 3 significant figures. (2 marks)

Cylinder volume is the area of the circle times the height: πr² h. Check that you have the radius, not the diameter.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Cylinders and Spheres

Area and volume

Lesson 6 of 7

Answer all questions. Show your working. · 30 minutes

Question 1 · 2 marks · Calculator

The diagram shows a cylinder with radius 5 cm and height 12 cm. Work out the volume of the cylinder. Give your answer correct to 3…

Question 2 · 3 marks · Calculator

Work out the total surface area of the cylinder in the last question. Give your answer correct to 3 significant figures.

Question 3 · 2 marks · Calculator

A sphere has a radius of 6 cm. Work out the volume of the sphere. Give your answer correct to 3 significant figures.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Cylinders and Spheres (cont.)

Area and volume

Lesson 6 of 7

Question 4 · 2 marks · Calculator

Work out the surface area of the sphere in the last question. Give your answer correct to 3 significant figures.

Question 5 · 3 marks · Calculator

A cylinder has a volume of 200 cm³ and a radius of 4 cm. Work out the height of the cylinder. Give your answer correct to 2 decimal places.

Question 6 · 2 marks · Non-calculator

A solid hemisphere has a radius of 3 cm. Work out the volume of the hemisphere. Give your answer in terms of π.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 2 marks · Calculator

Area and volume

Lesson 6 of 7

The diagram shows a cylinder with radius 5 cm and height 12 cm. Work out the volume of the cylinder. Give your answer correct to 3 significant figures. (2 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Area and volume

Lesson 6 of 7

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

π × 5² × 12

M1

942

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 3 marks · Calculator

Area and volume

Lesson 6 of 7

“

Work out the total surface area of the cylinder in the last question. 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 6 of 7

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

2πr²

M1

2πr h

M1

534

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 2 marks · Calculator

Area and volume

Lesson 6 of 7

“

A sphere has a radius of 6 cm. Work out the volume of the sphere. 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 3 · mark scheme

Area and volume

Lesson 6 of 7

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

4/3π × 6³

M1

905

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 2 marks · Calculator

Area and volume

Lesson 6 of 7

“

Work out the surface area of the sphere 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 6 of 7

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

4π × 6²

M1

452

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 3 marks · Calculator

Area and volume

Lesson 6 of 7

“

A cylinder has a volume of 200 cm³ and a radius of 4 cm. Work out the height of the cylinder. Give your answer correct to 2 decimal places.

HOW TO ANSWER IT

Command word: 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 6 of 7

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

π × 4² × h = 200

M1

200/16π

M1

3.98

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 2 marks · Non-calculator

Area and volume

Lesson 6 of 7

“

A solid hemisphere has a radius of 3 cm. Work out the volume of the hemisphere. Give your answer in terms of π.

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 6 of 7

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

⅔π × 27 or half of 4/3π × 27

M1

18π

A1

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