Solving problems in 3D.pptx

EDEXCEL GCSE MATHS · HIGHER

Solving problems in 3D

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Lesson 7 of 9

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EDEXCEL GCSE MATHS · HIGHER

Warm-up

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Lesson 7 of 9

Answer each one, then check.

1

What is Pythagoras' theorem?

2

What is tan θ in a right-angled triangle?

3

What is a diagonal?

4

What is the volume of a cuboid?

5

Work out √169.

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Warm-up - Answers

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Lesson 7 of 9

1

What is Pythagoras' theorem?

a² + b² = c²

2

What is tan θ in a right-angled triangle?

opposite ÷ adjacent

3

What is a diagonal?

A line joining two corners that are not next to each other

4

What is the volume of a cuboid?

l × w × h

5

Work out √169.

13

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Learning Objectives

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Lesson 7 of 9

1

Find the length of a diagonal of a cuboid.

2

Find the height of a pyramid or a slant edge using Pythagoras.

3

Find the angle between a line and a plane.

4

Draw the right-angled triangle you need out of a 3D shape.

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Lesson 7 of 9

3D PROBLEMS

To solve a 3D problem, find a right-angled triangle inside the shape, draw it separately, then use Pythagoras or trigonometry.

Sketch the flat triangle with its lengths and angle, and work in that triangle.

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A Diagonal in a Cuboid

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Lesson 7 of 9

Triangle ACG is right-angled at C. Find AC first, then AG.

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Diagonal of a Cuboid

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Lesson 7 of 9

Two steps, both using Pythagoras.

1

Base diagonal

AC² = l² + w²

2

Space diagonal

AG² = AC² + h² = l² + w² + h²

3

Angle with the base

tan θ= h/AC

4

Draw it

Sketch triangle ACG separately with its right angle at C

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Space Diagonal of a Cuboid

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Lesson 7 of 9

A cuboid measures 12 cm by 4 cm by 3 cm. Find the length of its longest diagonal.

1

Formula

d² = 12² + 4² + 3²

2

Add up

d² = 144 + 16 + 9 = 169

3

Square root

d = 13

ANSWER

The longest diagonal is 13 cm.

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Angle Between a Line and a Plane

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Lesson 7 of 9

A cuboid ABCDEFGH has AB = 8 cm, BC = 6 cm and CG = 5 cm. Find the angle between AG and the base ABCD.

1

Base diagonal

AC = √(8² + 6²) = 10

2

Right-angled triangle ACG

tan θ= CG/AC = 5/10

3

Inverse tangent

θ= tan ⁻¹(0.5) = 26.6°

ANSWER

The angle between AG and the base is 26.6° (1 d.p.).

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A Square-Based Pyramid

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Lesson 7 of 9

A pyramid has a square base of side 8 cm and a vertical height of 10 cm. The apex is directly above the centre of the base. Find the length of a sloping edge.

1

Half the base diagonal

√(8² + 8²)/2 = √128/2 = 5.657

2

Triangle with the height

edge² = 10² + 5.657² = 132

3

Square root

edge = 11.5

ANSWER

The sloping edge is 11.5 cm (3 s.f.).

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The Angle Between a Line and a Plane

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Lesson 7 of 9

Always use the right angle.

Drop a perpendicular

From the top of the line straight down to the plane.

Join to the base

Draw the line from the foot of the perpendicular to the bottom of the original line.

The angle

This is the angle between the line and the plane, and the triangle formed has a right angle.

Then

Use sin, cos or tan as usual.

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Key Terms

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Lesson 7 of 9

Cuboid

A 3D shape with six rectangular faces.

Space diagonal

A diagonal joining opposite corners through the inside of a cuboid.

Plane

A flat surface, such as the base.

Apex

The top point of a pyramid.

Slant edge

An edge from the apex to a base corner.

Angle of elevation

The angle looking up from horizontal.

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Lesson 7 of 9

YOUR TASK

Find the Triangle

12 minutes

A cuboid is 9 cm long, 12 cm wide and 8 cm high. (a) Find the base diagonal. (b) Find the space diagonal. (c) Find the angle between the space diagonal and the base.

1

Work in the base first.

2

Then use the height.

WHAT A GOOD ANSWER SHOWS

(a) √(9² + 12²) = 15 cm. (b) √(15² + 8²) = 17 cm. (c) tan ⁻¹(8 ÷ 15) = 28.1°.

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Can I...?

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Lesson 7 of 9

Draw a right-angled triangle from a 3D shape.

Find a base diagonal.

Find a space diagonal.

Use Pythagoras twice.

Find a pyramid slant edge.

Find an angle between a line and a plane.

Use tangent, sine or cosine.

Give sensible accuracy.

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Summary & Exam Focus

Space diagonal: d² = l² + w² + h².

Angle with the base: tan θ= height/(base diagonal).

Pyramid: use half the base diagonal with the height.

Always sketch the triangle you are using.

EXAM FOCUS

ABCDEFGH is a cuboid with AB = 8 cm, BC = 6 cm and CG = 5 cm. Work out the angle between AG and the base ABCD. Give your answer correct to 1 decimal place. (4 marks)

Find the base diagonal first, then draw the right-angled triangle with the height.

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