Right-angled trigonometry in 2D and 3D
A clear starting point, worked steps and space to practise.
Start here
In a 3D trigonometry problem, first identify the particular flat right triangle containing the angle you need.
The angle with a plane is measured to the diagonal’s projection on that plane, not automatically to one edge.
1 Work through these steps
- Sketch the solid and the relevant triangle.
- Use Pythagoras to find a face diagonal if necessary.
- Label opposite, adjacent and hypotenuse for the chosen angle.
- Use a trig ratio and keep intermediate values exact.
2 Follow a worked example
A rectangular box has a 3 m by 4 m base and height 5 m. What angle does its space diagonal make with the base?
- The base diagonal is √(3² + 4²) = 5 m.
- The right triangle has horizontal adjacent side 5 m and vertical opposite side 5 m.
- tan θ = 5/5 = 1, so θ = 45°.
3 Try it yourself
Show each step on paper or open Working out. Then check the answer.
-
A ramp rises 2 m over a horizontal run of 2 m. Find its angle to the horizontal.
Check answer 1
Answer: 45°
tan θ = 2/2 = 1, so θ = 45°.
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Working out
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Check your understanding
Explain the worked example in your own words. Say why the method works, not just what the answer is.
Skills and explanation links
Key ideas taught here
- In a 3D trigonometry problem, first identify the particular flat right triangle containing the angle you need.
- The angle with a plane is measured to the diagonal’s projection on that plane, not automatically to one edge.
Method and checking
Optional recap: related Maths skills
These earlier examples revisit background skills. Start with the topic example and practice above.
Useful vocabulary
- hypotenuse
- The longest side of a right-angled triangle.
- opposite
- The side across from the chosen angle.
- adjacent
- The side next to the chosen angle, not the hypotenuse.
- ratio
- A comparison used in sine, cosine and tangent.
Rules for these related skills
These side-ratio definitions apply to acute angles in right-angled triangles; trigonometric functions extend beyond this model.
Opposite and adjacent depend on the chosen angle.
Use degree mode for school angle questions unless told otherwise.
In a right triangle, identify the hypotenuse.
- The hypotenuse is opposite the right angle.
- It is always the longest side.
If opposite = 6 and hypotenuse = 10, find sin(theta).
- sin(theta) = opposite / hypotenuse.
- sin(theta) = 6/10 = 0.6.
Find x if tan(35 degrees) = x/12. Give the answer to two decimal places.
- Multiply both sides by 12.
- x = 12 x tan(35 degrees).
- In degree mode, x is approximately 8.40249, so x = 8.40 to two decimal places.
A ladder leans against a wall, making a right triangle. Which maths helps find the height?
- Right-angled trigonometry connects sides and angles.
- Choose sin, cos or tan based on the known information.
More visual models and everyday examples
Each everyday example has its own information and model. Use its stated values rather than copying values from another question.
Concept example. The story examples below have their own diagrams.
Label sides from the chosen angle.
For parents and teachers
- Ask your child to explain the method aloud before writing the answer.
- Use a real-life context at home, such as shopping, cooking, sport scores, maps or timetables.
- Praise clear working and checking, not only speed.
- Ask your child to write the formula or rule first, then substitute values carefully.
Practise choosing a method from the information in the question. The examples here are Pi Leo Academy learning activities. Official NAPLAN testing takes place in Years 3, 5, 7 and 9; a topic appearing in these notes does not mean it is tested in NAPLAN.
These notes provide guided examples and practice; they do not certify completion of every curriculum requirement.