Three-dimensional triangles and line–plane angles
Project a sloping line onto a plane and solve the right triangle formed by its vertical rise and horizontal projection.
Extended only.
Before you begin
- Use Pythagoras and right-triangle ratios.
- Identify perpendicular vertical and horizontal directions.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which line defines a sloping line's angle with a plane?
What you will learn
- Use Pythagoras in successive triangles in a solid.
- Find the angle between a line and a plane using its projection.
Build the floor triangle before the space triangle
For a cuboid with horizontal edges 3 and 4 cm and height 12 cm, the floor diagonal is √(3² + 4²) = 5 cm. The space diagonal then forms a right triangle with that floor diagonal and the vertical height, giving √(5² + 12²) = 13 cm.
A perspective drawing does not show all right angles at ninety degrees on the page. Identify the plane and draw each relevant triangle separately. A space diagonal is not the same as a face diagonal, and using just one floor edge misses the other horizontal component.
The line's projection defines its angle with a plane
The angle between the space diagonal and the horizontal plane is the angle with its perpendicular projection onto that plane. In the cuboid example, tan θ = 12/5, so θ ≈ 67.4°. It is not the angle with an arbitrary edge on the floor.
For a pyramid with apex directly above the centre, locate the centre-to-corner horizontal distance first, then combine it with the vertical height. Mark the right angle at the foot of the perpendicular. Keep exact roots or full calculator values through the two stages, and round the final requested length or angle only.
Pause and explain
A cuboid has floor edges 3 and 4, height 12. What is the space diagonal?
Worked example
A cuboid is 3 by 4 cm horizontally and 12 cm tall. Find its space diagonal and its angle to the base.
Show the worked solution
- The projection on the base is √(3² + 4²) = 5 cm.
- The space diagonal is √(5² + 12²) = 13 cm.
- tan θ = 12/5, so θ ≈ 67.4° to the base plane.
Answer Diagonal 13 cm; angle 67.4°.
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.
Find the space diagonal of a cuboid with dimensions 2, 3 and 6 cm.
Give me a hint
Use the floor diagonal squared without rounding its root.
Compare my reasoning
- The floor diagonal squared is 4 + 9 = 13.
- The space diagonal squared is 13 + 36 = 49.
- The diagonal is 7 cm.
7 cm.
Look for these in your work
- Included all three perpendicular dimensions.
- Retained exact intermediate values.
A box has floor edges 6 and 8 cm and height 5 cm. Find the space diagonal's angle to its floor.
Give me a hint
Its floor projection is the floor diagonal.
Compare my reasoning
- The projection has length √(36 + 64) = 10 cm.
- tan θ = 5/10.
- θ ≈ 26.6° to one decimal place.
26.6°.
Look for these in your work
- Used the full horizontal projection.
- Identified the requested plane angle.
A square pyramid has base side 8 cm and vertical height 6 cm, with apex above the centre. Find an apex-to-corner edge exactly.
Give me a hint
The horizontal projection is half a base diagonal.
Compare my reasoning
- The centre-to-corner distance is √(4² + 4²) = 4√2 cm.
- The edge squared is 32 + 36 = 68.
- The sloping edge is 2√17 cm; it differs from the triangular face's slant height.
2√17 cm.
Look for these in your work
- Located the base centre correctly.
- Distinguished corner edge from face height.
Common mistakes
- Confusing a face and space diagonal.
- Using a floor edge instead of a projection.
- Reading right angles from a perspective sketch.
What can you explain now?
Why do successive calculations retain unrounded values?
Compare with the explanation
Early rounding changes the second triangle's dimensions.
Use exact roots or stored calculator values until the final requested answer.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, redraw a solid's two calculation triangles and mark the projection defining its plane angle.
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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
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