Spatial right sections
Identify the projection before choosing a ratio
- Projection Join the line endpoint to the perpendicular foot in the plane
- Line–plane angle The acute angle between the line and its projection
- Calculation Find the base diagonal first and retain unrounded values
Worked example
A cuboid has a horizontal rectangular base with sides 6 m and 8 m and diagonal AC. D is 24 m vertically above C. Find the spatial diagonal AD and the tangent of its angle to the base plane
\[AC=\sqrt{6^2+8^2}=10\]
- \(\displaystyle AD=\sqrt{10^2+24^2}=26\)The vertical edge is perpendicular to the horizontal diagonal
- \(\displaystyle \tan\theta=\frac{24}{10}=\frac{12}{5}\)Use the complete horizontal projection
Common mistakes
Wrong projection
✕Tangent of the angle equals the height divided by one base edge✓Tangent of the angle equals the height divided by the full horizontal projection
The projection belongs to the section containing the spatial lineDifficulty levels
introBase diagonal and angle
standardSpatial brace and angle
challengeLengths, angle and a practical comparison
examRecognise the correct angle
Questions about this topic
Is a line–plane angle an angle between planes?+
No. A plane–plane angle needs two in-plane lines perpendicular to their common edge
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Specialist Mathematics, chapter 7: Trigonometry and transformations
In this chapter
Convert angle measure and use the radian arc relationArea of a sectorUse the sine rule in a non-right-angled triangleUse the cosine rule in a non-right-angled triangleTrigonometry in three dimensionsExact circular-segment area and perimeterChoose perpendicular sections and calculate spatial anglesUse exact trigonometric identities and quadrant informationCompound, double-angle and reciprocal identitiesTransform and reconstruct sine and cosine functionsCombine waves and convert trigonometric sums and productsMap or recover points and finite coordinate setsCompose two plane transformations in orderRotate, reflect and interpret fixed points and area