Right-triangle sections in solids
Project the line into the reference plane, then use the perpendicular section. The diagram is a representation of the geometry, not a source of measured angles
- Project into the plane Drop the top endpoint vertically onto the horizontal plane. Join that foot to the starting point. This is the line’s horizontal projection
- Find the horizontal distance For a rectangular base, the projection is the whole base diagonal, not one edge \(\displaystyle f=\sqrt{a^2+b^2}\)
- Use the perpendicular section The vertical height h is opposite the angle to the base plane. The horizontal projection f is adjacent \(\displaystyle \tan\theta=\frac{h}{f}\)
- Recover lengths when an angle is known Use the side pair in the section. Keep the full angle and intermediate lengths \(\displaystyle \begin{aligned}h&=f\tan\theta\\d&=\frac{f}{\cos\theta}\end{aligned}\)
- Check the reference The acute angle to the vertical is complementary to the angle to the horizontal plane. An oblique drawing does not preserve the apparent angles
Worked example
A cuboid has a 6 m by 8 m horizontal base and 6 m height. Find the angle of its space diagonal to the base plane, to one decimal place
- \(\displaystyle f=\sqrt{6^2+8^2}=10\text{ m}\)Find the full horizontal projection
- \(\displaystyle \tan\theta=\frac{6}{10}\)Use the perpendicular section, with height opposite and base diagonal adjacent
- \(\displaystyle \theta=\tan^{-1}(6/10)\)Use inverse tangent in degree mode
- \(\displaystyle \theta\approx30.9637565^\circ\)Keep the full angle before rounding
- \(\displaystyle \theta\approx31.0^\circ\)Round only the requested final angle
Common mistakes
Difficulty levels
Questions about this topic
Why is the base diagonal used?+
It joins the starting point to the vertical foot of the top endpoint, so it is the horizontal projection of the brace
Can I use sine or cosine instead?+
Yes, if the available side pair matches. For the angle to the base plane, sine is the height divided by the brace length, and cosine is the horizontal projection divided by the brace length
Can I measure the angle on the solid drawing?+
No. An oblique projection does not preserve the true angles. Use the measurements and the right-triangle section
Are pyramid corner and midpoint angles equal?+
No. At a fixed height, the shorter centre-to-midpoint projection gives a larger angle than the centre-to-corner projection
Do I need the sine rule or cosine rule?+
No. This section uses Pythagoras and right-triangle trigonometric ratios only
Does this prove a real brace will fit?+
No. Check actual internal fixing positions, measurement uncertainty, tolerances and the idealised perpendicular geometry
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 9: Pythagoras and trigonometry