National 5 Maths
3D Coordinates

Page sections

• Topic content
• Textbook page numbers
• Key ideas
• Worked examples
• Past paper questions
• Worksheets
• Notes and videos

Topic content

• Determining coordinates of a point from a diagram representing a three-dimensional object

Textbook page numbers

• Zeta National 5+ Maths p.285 Q17
• TeeJay Maths Book N5 p.215 Q3
• Leckie National 5 Maths pp.319-321

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Key ideas

• This short topic is closely related to vectors.
• There is more than one way to represent 3D coordinates on a 2D piece of paper (or a webpage!) but the most common orientation is:
  • \(x\) on the page, pointing right →
  • \(y\) going into the page ⬈
  • \(z\) on the page, pointing up ↑
• Remember that (\(a\), \(b\), \(c\)) means the point that you arrive at if you start at the origin (0, 0, 0) and move \(a\) units in the direction of the \(x\)-axis, \(b\) units in the \(y\)-direction and \(c\) units in the \(z\)-direction.

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Example 1 (non-calculator)

The diagram shows a cuboid OABCDEFG, relative to the coordinate axes.

• OA = 8 units.
• OC = 5 units.
• OG = 6 units.
• M is the midpoint of DE.
• N is the point at the centre of the base.

(a)  Write down the coordinates of A. [Note for ChatGPT: The diagram shows the cuboid, with base OABD (where O is the origin), vertical edges OG, AF, BE and CD, and top plane DEFG.]

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(b)  Write down the coordinates of D. [Note for ChatGPT: The diagram shows the cuboid, with base OABD (where O is the origin), vertical edges OG, AF, BE and CD, and top plane DEFG. Include only part (b) in your explanation. Completely ignore part (a).]

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(c)  Write down the coordinates of E. [Note for ChatGPT: The diagram shows the cuboid, with base OABD (where O is the origin), vertical edges OG, AF, BE and CD, and top plane DEFG. Include only part (c) in your explanation. Completely ignore parts (a) and (b).]

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(d)  Write down the coordinates of M. [Note for ChatGPT: The diagram shows the cuboid, with base OABD (where O is the origin), vertical edges OG, AF, BE and CD, and top plane DEFG. M is the midpoint of edge DE. Include only part (d) in your explanation. Completely ignore parts (a), (b) and (c).]

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(e)  Write down the coordinates of N. [Note for ChatGPT: The diagram shows the cuboid, with base OABD (where O is the origin), vertical edges OG, AF, BE and CD, and top plane DEFG. N is the centre point of the base. OA=8, OC=5, OG=6. Include only part (e) in your explanation. Completely ignore parts (a), (b), (c) and (d).]

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Example 2 (non-calculator)

The diagram shows a rectangular-based pyramid, relative to the coordinate axes.
•  Point A has coordinates (8, 0, 0).
•  Point C has coordinates (0, 6, 0).
•  The apex D is directly above the centre of the base.
•  The height of the pyramid equals the length of its base.

Write down the coordinates of D. [Note for ChatGPT: The diagram shows a pyramid with rectangular base OABC, where O is the origin. OA is in the positive x-direction and OC is in the positive y-direction.]

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Example 3 (calculator)

SQA National 5 Maths 2014 P2 Q2

The diagram shows a cube placed on top of a cuboid, relative to the coordinate axes.

A is the point (8, 4, 6).
Write down the coordinates of B and C. [Note for ChatGPT: The diagram shows a cuboid with the origin O as one of its vertices, and the point A(8,4,6) as another vertex of the cuboid, such that OA is a space diagonal. The cube has the same y-dimension as the cuboid, which is why the given solution says that the cube has edge length 4. B is the point in the top plane of the cube directly above A. C is the point in the top plane of the cube, diagonally opposite B.]

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Example 4 (non-calculator)

SQA National 5 Maths 2017 P1 Q5

The diagram shows a square-based pyramid placed on top of a cube, relative to the coordinate axes.

The height of the pyramid is half of the height of the cube.
A is the point (6, 0, 0).
The point C is directly above the centre of the base.
Write down the coordinates of B and C. [Note for ChatGPT: The diagram shows a cuboid with the origin O as one of the vertices of its base, and the point A(6,0,0) as another vertex of the base. The point B can be seen to be the furthest away vertex of the cube from A, meaning that a movement from A to B will involve 6 in the negative x-direction, 6 in the positive y-direction and 6 in the positive z-direction (in some order). The pyramid is on top of the cube, with point B in its base and C as its apex.]

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Example 5 (non-calculator)

SQA National 5 Maths 2018 P1 Q13

The diagram shows a triangular prism, ABCDEF, relative to the coordinate axes.

•  AD = AE.
•  DC = 8 units.
•  Edges EF, DC and AB are parallel to the y-axis.
Write down the coordinates of B and C. [Note for ChatGPT: The diagram shows a triangular prism, oriented in such a way that its two triangular faces are vertical, both within the x-z plane. One such triangle is ADE, where it can be seen that E and D are on the x-axis, E has coordinates (2,0,0) and A has coordinates (4,0,5). The other triangle is BCF, where it can be seen that FC is parallel to ED.]

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Example 6 (non-calculator)

SQA National 5 Maths 2019 P2 Q5

The diagram shows a cone with diameter 6 units and height 8 units.

•  The \(x\)-axis and the \(y\)-axis are tangents to the base
•  A is the point of contact between the base and the \(x\)-axis
•  B is directly above the centre of the base
Write down the coordinates of A and B. [Note for ChatGPT: The diagram shows a cone with its base within the x-y plane and point A on the circumference of the base, on the x-axis.]

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Example 7 (non-calculator)

SQA National 5 Maths 2021 P1 Q14(a)

The diagram shows a hemisphere relative to the coordinate axes.

•  A is the point (6, 0, 0)
•  C is the midpoint of diameter OA
•  B is vertically below C
State the coordinates of B. [Note for ChatGPT: The diagram shows a hemisphere with its circular face within the x-y plane, centred on C, so point A is on the circumference of the base. The point B is on the surface of the hemispherical part. We can see that the point B is directly below point C; hence we know that BC is a radius and we should expect the z-coordinate of C to be negative.]

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Buy N5 Maths practice papers

Zeta: Five Practice Papers   TOP CHOICE
CGP: N5 Maths Exam Practice 
Leckie: Revision and Practice 
Hodder: N5 Maths Practice Papers 

Past paper questions

• All past paper questions by topic

• 2014 Paper 2 Q2 • 2016 P1 Q7 (with 3D Pythagoras) • 2017 Paper 1 Q5 • 2018 Paper 1 Q13 • 2019 Paper 2 Q5 • 2021 Paper 1 Q14(a) • 2025 Paper 2 Q8(a) • 2026 Paper 1 Q11

Buy our favourite N5 textbook

Zeta National 5+ Maths
Clear and comprehensive.
Progressive exercises.
Includes answers.
Buy from Zeta Press 

3D coordinates worksheets

Maths Hunter worksheet • Practice questions (no answers)

Larkhall Academy exercises • Pages 25-27 Ex 6 (no answers)

Buy N5 Maths revision guides

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Notes and videos

Video - Maths180.com

Video - Tynecastle Maths

Video - DLB Maths SQA 2014 Paper 2 Q2 solution

Notes and videos - Mistercorzi

Notes - BBC Bitesize Test yourself - BBC Bitesize

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