Quick revision notes ยท Mathematics 4024 ยท Chapter 33 of 35
3D Trigonometry
Pythagoras and trigonometry in cuboids, pyramids and prisms, and the angle between a line and a plane.
At a glance
Pythagoras and trigonometry in cuboids, pyramids and prisms, and the angle between a line and a plane.
1Lengths in 3D
- Find a right-angled triangle that contains the length you want, then use Pythagoras.
- Usually two steps: the diagonal of the base first, then the vertical triangle.
- Space diagonal of a cuboid a ร b ร c: โ(a2 + b2 + c2).
Formulas
Worked example
A box is 12 cm long, 4 cm wide and 3 cm high. Find the length of the longest rod that fits inside.
- Base diagonal: โ(144 + 16) = โ160.
- Space diagonal: โ(160 + 9) = โ169.
13 cm
2The angle between a line and a plane
- Drop a perpendicular from the top of the line to the plane; the foot gives the line's projection.
- The angle is between the line and its projection โ not the vertical.
- The right-angled triangle is: line (hypotenuse), projection (adjacent), vertical height (opposite).
How to do it
- Identify the line and the plane.
- Find the projection of the line on the plane.
- Draw the right-angled triangle flat on its own.
- Use SOH CAH TOA.
Worked example
A cuboid is 6 cm by 8 cm by 5 cm high. Find the angle between the space diagonal and the base.
- Base diagonal (projection) = 10 cm.
- tan ฮธ = 5 รท 10.
26.6ยฐ
3Pyramids
- The vertical height goes from the apex to the centre of the base.
- For a square or rectangular base, the centre is the midpoint of the base diagonal.
- The angle between a sloping edge and the base uses half the base diagonal; the angle between a sloping face and the base uses half the base width.
Worked example
A square-based pyramid has base 8 cm ร 8 cm and vertical height 6 cm. Find the angle between a sloping edge and the base.
- Half the base diagonal = 4โ2 = 5.657 cm.
- tan ฮธ = 6 รท 5.657.
46.7ยฐ
4Prisms and wedges
- Look for the right angles between the faces.
- Redraw each triangle you need as a flat 2D triangle with its lengths.
- You may need the sine or cosine rule in a non-right-angled face.
Worked example
A wedge has a horizontal base 12 cm by 5 cm and a vertical back edge 9 cm high above one 5 cm side. Find the length from a front bottom corner to the opposite top corner.
- Base diagonal: โ(144 + 25) = 13.
- Then with the 9 cm height: โ(169 + 81) = โ250.
15.8 cm (3 s.f.)
โCommon mistakes
| Donโt write | Write instead |
|---|---|
| Using the base side instead of the base diagonal | The projection of a space diagonal is the base diagonal |
| Measuring the angle from the vertical | The angle with a plane is measured from the plane |
| Rounding โ160 to 12.6 early | Keep it exact or keep all digits |
โ In the exam
- Draw every triangle you use separately and label it.
- Name the angle with letters (e.g. angle AGE) so the examiner can follow.
- Give angles to 1 decimal place.