Solve three-dimensional trigonometry problems
Build right triangles inside a solid to find lengths, angles and the angle between a line and a plane.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
A cuboid's space diagonal passes through how many dimensions?
What you will learn
- Use Pythagoras to find face and space diagonals
- Apply right-triangle trigonometry to a three-dimensional diagram
- Find the angle between a line and a plane using the line's projection
Create a two-dimensional cross-section
A face diagonal lies on one rectangular face. A space diagonal spans the solid. Use Pythagoras on a face first, then combine the face diagonal with the remaining perpendicular dimension.
Mark which sides and angles are in the same right triangle. A line through a solid is not automatically a side of the triangle you need.
Measure a line against its projection
The angle between a line and a plane is the acute angle between the line and its perpendicular projection onto the plane. Join the line's endpoint to the foot of its perpendicular on the plane to identify the projection.
That line, its projection and the perpendicular height form a right triangle. Choose sine, cosine or tangent from the labelled angle and round only the final answer.
Pause and explain
How do you find the angle between a line and a plane?
Worked example
A cuboid is 6 cm by 8 cm by 12 cm. Find the angle between its space diagonal from a bottom corner and the base plane.
Show the worked solution
- The projection of the space diagonal onto the base is the base diagonal: √(6² + 8²) = 10 cm.
- The vertical height is 12 cm, so tan θ = 12/10.
- θ = tan⁻¹(1.2) ≈ 50.2°.
Answer 50.2°.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A cuboid has a 5 cm by 12 cm base and height 9 cm. Find its space diagonal.
Give me a hint
Find a base diagonal first, then use that with the vertical height.
Compare my reasoning
- Base diagonal = √(5² + 12²) = 13 cm.
- Space diagonal² = 13² + 9² = 169 + 81 = 250.
- Space diagonal = √250 = 5√10 cm.
5√10 cm.
Look for these in your work
- Base diagonal calculated
- Vertical dimension included
- Exact length simplified
A cuboid has a base 9 cm by 12 cm and height 8 cm. Find the angle between a space diagonal and the base plane to 1 decimal place.
Give me a hint
The base diagonal is the projection; the height is perpendicular to the base.
Compare my reasoning
- Projection length = √(9² + 12²) = 15 cm.
- tan θ = 8/15.
- θ = tan⁻¹(8/15) ≈ 28.1°.
28.1°.
Look for these in your work
- Projection identified
- Tangent relation formed
- Angle rounded to 1 d.p.
A wire runs from one corner of a 3 m by 4 m by 12 m room to the opposite ceiling corner. Find its length and angle above the floor to 1 decimal place.
Give me a hint
Project the wire onto the floor, then use the floor diagonal and vertical height.
Compare my reasoning
- Floor projection = √(3² + 4²) = 5 m.
- Wire length = √(5² + 12²) = 13 m.
- Angle above floor: tan θ = 12/5, so θ ≈ 67.4°.
Wire length 13 m; angle 67.4° above the floor.
Look for these in your work
- Floor projection calculated
- Space length found
- Angle tied to the floor plane
Common mistakes
- Comparing a line with a plane edge rather than its projection
- Using a face diagonal as if it were the space diagonal
- Drawing an angle outside the right triangle
- Rounding intermediate diagonal lengths
What can you explain now?
What is the projection used to define the angle between a line and a plane?
Compare with the explanation
The line's perpendicular projection onto the plane
The angle is measured between the original line and that projected segment.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why must you draw the perpendicular projection before measuring a line's angle to a plane?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.