Right triangles in three-dimensional shapes · Higher
| English | 中文 | Pinyin |
|---|---|---|
| projection/prəˈdʒekʃn/ | 投影 | tóu yǐng |
A cuboid’s space diagonal is longer than its base diagonal. Two different right triangles are needed to connect its width, depth and height.
- A cuboid’s space diagonal is longer than its base diagonal. Two different right triangles are needed to connect its width, depth and height.
- This lesson studies projection 投影: The view or component of a spatial length on a selected plane.
Choose the mathematical structure
- Identify a plane containing the wanted length or angle. Calculate a base diagonal first, then combine it with the perpendicular height. For an angle between a line and a plane, use the angle between the line and its orthogonal projection onto that plane. Mark right angles; general-triangle rules are used only when the selected triangle is not right-angled.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines projection?
The view or component of a spatial length on a selected plane.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
A cuboid of width 3, depth 4 and height 12 has base diagonal √(9+16)=5 and space diagonal √(25+144)=13. The angle α of the space diagonal to the horizontal base satisfies tanα=12/5, giving about 67.4°. Its sine is 12/13; its cosine is 5/13. In a square-based pyramid of side 6 and vertical height 4, the base centre-to-side-midpoint distance is 3, giving face slant height 5. The centre-to-corner distance is 3√2, giving edge length √(18+16)=√34. The face slant and edge lengths differ because their base projections differ.
Right triangles in three-dimensional shapes
Identify a plane containing the wanted length or angle
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find the base diagonal of a 3×4×12 cuboid.
√(3²+4²)=5.
Test a tempting shortcut
- Do not combine unrelated lengths as if they met at a right angle. The angle to a plane uses the base projection, not an arbitrary base edge. A pyramid’s face slant is different from its sloping edge.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A cuboid’s space diagonal can always be found from just two of its three edge lengths. This claim is false. Explain which definition or assumption it violates.
Find its space diagonal.
√(5²+12²)=13.
A cuboid’s space diagonal can always be found from just two of its three edge lengths.
Do not combine unrelated lengths as if they met at a right angle. The angle to a plane uses the base projection, not an arbitrary base edge. A pyramid’s face slant is different from its sloping edge.
Interpret a new situation
- AQA G20 Higher extends right-triangle and, where possible, general-triangle reasoning into 3D. Draw the relevant section triangle separately and name which spatial points it represents.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find tangent of the space-diagonal angle to the base.
Opposite 12 divided by base projection 5.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The view or component of a spatial length on a selected plane. Choose the relationship, show the method, check its assumptions and interpret the result.