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3D pythagoras / trig.

Free Printable 3D pythagoras / trig. PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 7)

Download free 3d pythagoras / trig. worksheets (PDF), GCSE maths past paper questions and worked solutions for 3D pythagoras / trig. (7Q). This is a Grade 7 geometry topic from the GCSE questions by topic collection for Higher tier GCSE Maths (Edexcel, AQA, OCR and IGCSE).

The free PDF worksheets table below brings together printable question booklets (Qs) and worked answers or mark schemes (Ans) from Maths Genie, Maths4Everyone, Corbett Maths, Maths Made Easy, 1st Class Maths, Maths Takeaway, Dr Austin, Maths DIY, Tired Old Cliche, E Classroom and PMT, along with past paper question booklets.

Below the worksheet links you can read our 3d pythagoras / trig. revision notes and worked examples with diagrams, try real GCSE past paper 3d pythagoras / trig. questions with step-by-step solutions, or download a PDF of all 3d pythagoras / trig. worksheet links.

Free 3D pythagoras / trig. PDF Worksheets & Answers

Grade 7 • Topic 7Q
Resource Questions Answers
GCSE past paper questions & solutions
 Qs Ans
Graded exam questions & solutions
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GCSE 9–1 practice booklet & solutions
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Practice Questions & answers
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Textbook Exercise & answers
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MME worksheet & model answers
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Edexcel exam questions & solutions
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AQA exam questions & solutions
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GCSE Maths Takeaway exam questions
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Practice grid / worksheet & answers
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Exam-style booklet & solutions
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Tired Old Cliche
3D pythagoras / trig. questions & answers
 Qs Ans
GCSE topic booklet & answers
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PMT (Edxl)
Edexcel GCSE topic questions & solutions
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PMT (AQA)
AQA GCSE topic questions & solutions
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PMT (IGCSE)
Edexcel IGCSE topic questions & solutions
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Online topic notes & examples
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Self-marking online worksheet
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Grade 7 topics
7A - Surds7B - Bounds
7C - Proportion (harder)7D - Algebraic fractions
7E - The quadratic formula7F - Factorising harder quadratics
7G - Trigonometric & exp. Graphs7H - Rearranging (harder)
7J - Functions (harder)7K - Iteration
7L - Area of any triangle7M - The sine rule
7N - The cosine rule7P - Congruent triangles
7Q - 3D pythagoras / trig.7R - Histograms
7S - Conditional probability7T - Exact trig. values (harder)
7U - Area of a segment
GCSE Maths Grade 7 Revision Notes & Method

Guide to 3D pythagoras / trig.: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

Explanation:

To solve 3D Pythagoras and trigonometry problems, identify and sketch the key 2D right-angled triangle inside the 3D shape. For a cuboid with dimensions l, w, and h, the floor diagonal is √(l2 + w2) and the 3D space diagonal can be found in one step using d = √(l2 + w2 + h2).

The angle between a line (such as a space diagonal or slant edge) and a horizontal plane (the base) is always the angle between the line and its projection (shadow) on the base directly underneath it.

Example 1: A cuboid ABCD.EFGH measures 12 cm by 9 cm by 8 cm (vertical height). Find space diagonal AG and its angle with the base ABCD (1 d.p.).

A B C G 12 cm 9 cm 8 cm

Solution:

Base diagonal AC = √(122 + 92) = √225 = 15 cm
Space diagonal AG = √(152 + 82) = √289 = 17 cm
In ΔACG: tan(∠GAC) = 815 ⇒ ∠GAC = 28.072...°
Answer: AG = 17 cm, angle = 28.1°

Example 2: A square-based pyramid has base side 10 cm and slant edges 13 cm. Find its vertical height h (3 s.f.).

h M √50 13 cm 10 cm

Solution:

Base diagonal = √(102 + 102) = √200 cm
Half diagonal (corner to M) = √2002 = √50 cm
h = √(132 − (√50)2) = √(169 − 50) = √119
Answer: h = 10.9 cm (3 s.f.)
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper 3D pythagoras / trig. Questions & Answers

Past Paper Example 1 GCSE Jun 2018 2H Q18 4 marks
Full Paper (PDF) • Mark Scheme (PDF)
ABCDEFGH is a cuboid.
FGEHABDC
AB = 7.3 cm
CH = 8.1 cm
Angle BCA = 48°
Find the size of the angle between AH and the plane ABCD.
Give your answer correct to 1 decimal place.
Worked Solution & Mark Scheme (4 marks)
In right-angled triangle ABC: sin 48° = 7.3 / AC ⇒ AC = 7.3 / sin 48° = 9.8231 cm M3
In right-angled triangle ACH: tan(CAH) = CH / AC = 8.1 / 9.8231 = 0.82459 ⇒ angle CAH = 39.5°. A1
Answer: 39.5
Past Paper Example 2 IGCSE Jun 2018 2H Q19 3 marks
Full Paper (PDF) • Mark Scheme (PDF)
The diagram shows a triangular prism.
ABCDEF10 cm24 cm8 cm
AF = 10 cm, AB = 24 cm and BC = 8 cm.
Angle FAB = angle ADC = angle BCD = 90°
Work out the size of the angle between the line BE and the plane ABCD.
Give your answer correct to 1 decimal place.
Worked Solution & Mark Scheme (3 marks)
BD² = 24² + 8² = 576 + 64 = 640 ⇒ BD = √640 = 25.298… cm M2
tan(∠DBE) = ED / BD = 10 / √640 = 0.39528… ⇒ ∠DBE = 21.568…° = 21.6° (1 d.p.) A1
Answer: 21.6 °
Past Paper Example 3 IGCSE Jan 2023 1HR Q21 4 marks
Full Paper (PDF) • Mark Scheme (PDF)
The diagram shows a triangular prism, ABCDEF, with a rectangular base ABCD
18°ABCDEF2.2 cm6 cmDiagram NOTaccurately drawn
AB = 6 cm   DE = 2.2 cm   angle DAE = 18°   angle ADE = 90°
Work out the angle that BE makes with the plane ABCD
Give your answer correct to one decimal place.
Worked Solution & Mark Scheme (4 marks)
In △ ADE: AD = 2.2\tan 18° = 6.7709 cm M2
In base ABCD: BD = √(62 + 6.77092) = 9.0468 cm M1
Angle DBE = \tan-1(2.29.0468) = 13.668° = 13.7° (1 d.p.). A1
Answer: 13.7 °

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