A three-dimensional triangle question is a set of flat triangles in disguise. Your job is to draw each one separately, then solve them in the right order.
This lesson is part of solution of triangles. It combines the sine rule with basic right-angled trigonometry.
How do I pull flat triangles out of the diagram?
Follow the same four steps every time.
- Redraw the ground as a flat triangle, and mark the foot of the pole or tower.
- Redraw the vertical plane as a right-angled triangle standing on one ground line.
- Write every given length and angle on the correct triangle.
- Solve the triangle that already has enough data, then pass the length it finds to the next.
The angle of elevation belongs in the vertical triangle only. Ground angles, such as angle PAB, belong in the horizontal triangle.
Worked example: a flagpole seen from two points
A flagpole PT stands vertically on level ground, with P at its foot and T at its top. Points A and B are on the ground with AB = 60 m, angle PAB = 40° and angle PBA = 75°. The angle of elevation of T from A is 25°. Find the height PT.
Step 1: the ground triangle ABP. Angle APB = 180° − 40° − 75° = 65°. AP is opposite B, and AB is opposite P, so the sine rule gives:
AP ÷ sin 75° = 60 ÷ sin 65°
AP = 60 × 0.9659 ÷ 0.9063 ≈ 63.95 m.
Step 2: the vertical triangle APT. The triangle is right-angled at P, with angle of elevation 25° at A:
PT = AP × tan 25° = 63.95 × 0.4663 ≈ 29.8 m.
The mistake that costs marks
The common slip is to use the elevation angle inside the ground triangle. A student sees “25°” and puts it in triangle ABP, which gives a length that depends on a wrong angle sum.
| Step | Wrong | Right |
|---|---|---|
| Place the 25° | In ABP with the ground angles | In the vertical triangle APT |
| Angles of ABP | 40°, 75°, 25° (sum 140°) | 40°, 75°, 65° (sum 180°) |
| Result | AP is wrong | AP ≈ 63.95 m, PT ≈ 29.8 m |
An easy self-check: the three angles of each flat triangle must add to 180°. If they do not, an angle is in the wrong triangle.
What if the question asks for a different length?
Questions may ask for the distance BT or the angle of elevation from B. The ground triangle still comes first, and you then use the vertical triangle at B.
For BT, find BP from the sine rule in ABP: BP ÷ sin 40° = 60 ÷ sin 65°, so BP ≈ 42.55 m. Then BT² = BP² + PT², so BT = √(42.55² + 29.82²) ≈ 51.96 m, and the elevation at B satisfies tan θ = 29.82 ÷ 42.55.
Keep four or five figures in the working so the final rounding stays accurate.
Check yourself
Points A and B are on level ground with AB = 40 m. A vertical pole PT has its foot at P. Angle PAB = 50° and angle PBA = 70°, and the angle of elevation of T from A is 30°. Find PT.
Answer
Angle APB = 180° − 50° − 70° = 60°.
AP ÷ sin 70° = 40 ÷ sin 60°, so AP = 40 × 0.9397 ÷ 0.8660 ≈ 43.40 m.
PT = AP tan 30° = 43.40 × 0.5774 ≈ 25.1 m.
What to study next
Test every skill in this chapter on mixed questions with the solution of triangles practice set. The word-problem structure worksheet helps you organise the flat triangles before you calculate.
If you want a teacher to work through 3D questions with you, see online one-to-one Additional Mathematics tuition.