Similarity in 3D solids - Teacher Slides.pptx

EDEXCEL GCSE MATHS · HIGHER

Similarity in 3D solids

Similarity and congruence

Lesson 5 of 5

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EDEXCEL GCSE MATHS · HIGHER

Warm-up

Similarity and congruence

Lesson 5 of 5

Answer each one, then check.

1

Work out 2³.

2

Work out 1.5³.

3

What is the area scale factor if k = 3?

4

Work out ∛27.

5

What is the formula for the volume of a cone?

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EDEXCEL GCSE MATHS · HIGHER

Warm-up - Answers

Similarity and congruence

Lesson 5 of 5

1

Work out 2³.

8

2

Work out 1.5³.

3.375

3

What is the area scale factor if k = 3?

9

4

Work out ∛27.

3

5

What is the formula for the volume of a cone?

⅓πr² h

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EDEXCEL GCSE MATHS · HIGHER

Learning Objectives

Similarity and congruence

Lesson 5 of 5

1

Use length, area and volume scale factors.

2

Find a volume, capacity or mass of a similar solid.

3

Work out the length ratio from a volume ratio.

4

Solve problems with similar cones, cylinders and frustums.

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EDEXCEL GCSE MATHS · HIGHER

Similar Cubes

Similarity and congruence

Lesson 5 of 5

Doubling lengths multiplies areas by 4 and volumes by 8.

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EDEXCEL GCSE MATHS · HIGHER

Scale Factors for Similar Solids

Similarity and congruence

Lesson 5 of 5

If the length scale factor is k.

Measure

Scale factor

If k = 2

Lengths

k

2

Areas (including surface area)

k²

4

Volumes (and capacity, mass)

k³

8

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EDEXCEL GCSE MATHS · HIGHER

Volume of a Similar Cone

Similarity and congruence

Lesson 5 of 5

Cone A has height 6 cm and volume 40 cm³. Cone B is similar to A and has height 9 cm. Find the volume of cone B.

1

Length scale factor

9/6 = 1.5

2

Volume scale factor

1.5³ = 3.375

3

Volume of B

40 × 3.375

ANSWER

135 cm³

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EDEXCEL GCSE MATHS · HIGHER

A Model and the Real Thing

Similarity and congruence

Lesson 5 of 5

A model statue is 20 cm tall and has mass 2 kg. The real statue is 1.2 m tall and made of the same material. Find the mass of the real statue.

1

Length scale factor

120 ÷ 20 = 6

2

Volume (and mass) scale factor

6³ = 216

3

Mass

2 × 216

ANSWER

432 kg

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EDEXCEL GCSE MATHS · HIGHER

A Frustum

Similarity and congruence

Lesson 5 of 5

A cone of height 15 cm has volume 500 cm³. A small cone of height 6 cm is cut off the top, leaving a frustum. Find the volume of the frustum.

1

The small cone is similar to the large cone

Length scale factor 6/15 = 0.4

2

Volume scale factor

0.4³ = 0.064

3

Volume of the small cone

500 × 0.064 = 32

4

Frustum = large cone minus small cone

500 − 32

ANSWER

468 cm³

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EDEXCEL GCSE MATHS · HIGHER

From Volumes to Lengths

Similarity and congruence

Lesson 5 of 5

Two similar solids have volumes 8 cm³ and 125 cm³. Find the ratio of their lengths and of their surface areas.

1

Take the cube root of the volumes

∛8 : ∛125 = 2 : 5

2

Lengths

2 : 5

3

Surface areas: square the ratio

4 : 25

ANSWER

Lengths 2 : 5; surface areas 4 : 25.

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EDEXCEL GCSE MATHS · HIGHER

Which Scale Factor?

Similarity and congruence

Lesson 5 of 5

USE K (LENGTHS)

• Heights, radii, edges, perimeters.

• Anything measured in a straight line.

• Read directly from the two shapes.

USE K² OR K³

• k²: area, surface area, cross-section.

• k³: volume, capacity, mass (same material).

• Cube or square root to go backwards.

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EDEXCEL GCSE MATHS · HIGHER

Key Terms

Similarity and congruence

Lesson 5 of 5

Similar solids

Solids with the same shape and all lengths in the same ratio.

Volume scale factor

k³.

Surface area scale factor

k².

Frustum

A cone or pyramid with the top cut off parallel to the base.

Capacity

The volume a container holds.

Density

Mass per unit volume; the same in similar solids of the same material.

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EDEXCEL GCSE MATHS · HIGHER

Similarity and congruence

Lesson 5 of 5

YOUR TASK

Bottle Sizes

12 minutes

A small bottle is 10 cm tall and holds 250 ml. A similar large bottle is 15 cm tall. Find the capacity of the large bottle. A label on the small bottle has area 40 cm²; find the label area on the large bottle.

1

Find k.

2

Cube for volume.

3

Square for area.

WHAT A GOOD ANSWER SHOWS

Scale factor 1.5. Capacity 250 × 1.5³ = 843.75 ml, about 844 ml. Label 40 × 1.5² = 90 cm².

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EDEXCEL GCSE MATHS · HIGHER

Can I...?

Similarity and congruence

Lesson 5 of 5

State the scale factors k, k², k³.

Find a volume of a similar solid.

Find a mass using a volume factor.

Find the surface area of a similar solid.

Use a cube root to find a length ratio.

Find the volume of a frustum.

Choose the right scale factor.

Show clear working.

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EDEXCEL GCSE MATHS · HIGHER

Summary & Exam Focus

Lengths k, areas k², volumes k³.

For the same material, mass scales like volume.

From a volume ratio take the cube root for lengths.

A frustum is a large cone minus a similar small cone.

EXAM FOCUS

Cone A has height 6 cm and volume 40 cm³. Cone B is similar to A and has height 9 cm. Work out the volume of cone B. (3 marks)

Find the length scale factor first. Cube it for volume. Do not square it.

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EDEXCEL GCSE MATHS · HIGHER

Exam Practice: Similarity in 3D Solids

Similarity and congruence

Lesson 5 of 5

Answer all questions. Show your working. · 30 minutes

Question 1 · 3 marks · Work out

Cone A and cone B are mathematically similar. The height of cone A is 6 cm and its volume is 40 cm³. The height of cone B is 9 cm. Work out…

Question 2 · 3 marks · Work out

A model of a statue is 20 cm tall and has a mass of 2 kg. The real statue is 1.2 m tall and is made from the same material. Work out the…

Question 3 · 4 marks · Work out

A cone has a height of 15 cm and a volume of 500 cm³. A smaller cone of height 6 cm is cut off the top, leaving a frustum. The small cone…

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EDEXCEL GCSE MATHS · HIGHER

Exam Practice: Similarity in 3D Solids (cont.)

Similarity and congruence

Lesson 5 of 5

Question 4 · 3 marks · Work out

Two similar solids have surface areas in the ratio 4 : 9. Work out the ratio of their volumes.

Question 5 · 3 marks · Work out

A small bottle is 10 cm tall and holds 250 ml of juice. A similar large bottle is 15 cm tall. Work out the capacity of the large bottle…

Question 6 · 3 marks · Work out

Two similar solids have volumes of 8 cm³ and 125 cm³. The surface area of the smaller solid is 20 cm². Work out the surface area of the…

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EDEXCEL GCSE MATHS · HIGHER

Question 1 · 3 marks · Work out

Similarity and congruence

Lesson 5 of 5

Cone A and cone B are mathematically similar. The height of cone A is 6 cm and its volume is 40 cm³. The height of cone B is 9 cm. Work out the volume of cone B. (3 marks)

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EDEXCEL GCSE MATHS · HIGHER

Question 1 · mark scheme

Similarity and congruence

Lesson 5 of 5

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

Length scale factor 1.5

M1

1.5³ = 3.375

M1

135

A1

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EDEXCEL GCSE MATHS · HIGHER

Question 2 · 3 marks · Work out

Similarity and congruence

Lesson 5 of 5

“

A model of a statue is 20 cm tall and has a mass of 2 kg. The real statue is 1.2 m tall and is made from the same material. Work out the mass of the real statue.

HOW TO ANSWER IT

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

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EDEXCEL GCSE MATHS · HIGHER

Question 2 · mark scheme

Similarity and congruence

Lesson 5 of 5

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

Scale factor 6

M1

6³ = 216

M1

432

A1

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EDEXCEL GCSE MATHS · HIGHER

Question 3 · 4 marks · Work out

Similarity and congruence

Lesson 5 of 5

“

A cone has a height of 15 cm and a volume of 500 cm³. A smaller cone of height 6 cm is cut off the top, leaving a frustum. The small cone is similar to the large cone. Work out the volume of the frustum.

HOW TO ANSWER IT

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

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EDEXCEL GCSE MATHS · HIGHER

Question 3 · mark scheme

Similarity and congruence

Lesson 5 of 5

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

Scale factor 0.4

M1

0.4³ = 0.064

M1

Small cone 32

A1

468

A1

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EDEXCEL GCSE MATHS · HIGHER

Question 4 · 3 marks · Work out

Similarity and congruence

Lesson 5 of 5

“

Two similar solids have surface areas in the ratio 4 : 9. Work out the ratio of their volumes.

HOW TO ANSWER IT

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

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EDEXCEL GCSE MATHS · HIGHER

Question 4 · mark scheme

Similarity and congruence

Lesson 5 of 5

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

Length ratio 2 : 3

M1

Cubing

M1

8 : 27

A1

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EDEXCEL GCSE MATHS · HIGHER

Question 5 · 3 marks · Work out

Similarity and congruence

Lesson 5 of 5

“

A small bottle is 10 cm tall and holds 250 ml of juice. A similar large bottle is 15 cm tall. Work out the capacity of the large bottle. Give your answer to the nearest ml.

HOW TO ANSWER IT

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

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EDEXCEL GCSE MATHS · HIGHER

Question 5 · mark scheme

Similarity and congruence

Lesson 5 of 5

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

Scale factor 1.5

M1

250 × 1.5³

M1

844

A1

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EDEXCEL GCSE MATHS · HIGHER

Question 6 · 3 marks · Work out

Similarity and congruence

Lesson 5 of 5

“

Two similar solids have volumes of 8 cm³ and 125 cm³. The surface area of the smaller solid is 20 cm². Work out the surface area of the larger solid.

HOW TO ANSWER IT

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

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EDEXCEL GCSE MATHS · HIGHER

Question 6 · mark scheme

Similarity and congruence

Lesson 5 of 5

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

Length scale factor 5/2

M1

Area scale factor 6.25

M1

125

A1

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