Measure elevation and depression from the horizontal and include eye height
Identify the horizontal angle reference · Use vertical difference rather than total height in the triangle · Relate elevation and depression under parallel horizontals
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Tangent and inverse ratios
- Parallel-line angle facts
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Identify the horizontal angle reference
- Use vertical difference rather than total height in the triangle
- Relate elevation and depression under parallel horizontals
Use the horizontal reference
Elevation is measured upward from the observer's horizontal to the sight line. Depression is measured downward from a horizontal at the higher observer. With parallel horizontals, the corresponding elevation and depression angles agree; neither is measured from the vertical.
Add back the observer height
For level ground, horizontal distance d, eye height e and elevation θ to a tower top, H=e+d tanθ. The triangle vertical leg is H−e, not H. The illustrated e=1.6 m, d=20 m, θ=35° gives H≈15.6042 m. An angle of 55° from the vertical corresponds to 35° from the horizontal. State the level-ground and vertical-tower assumptions.
Sources for this lesson
Pause and explain
Is an angle of depression measured from the vertical?
Worked example
Find the illustrated tower height to 3 significant figures.
Show the worked solution
- Vertical difference=20 tan35°≈14.004151 m.
- Add eye height 1.6 m.
Answer 15.6 m.
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
On level ground, an observer with eye height 1.5 m is 20 m horizontally from a vertical tower. Elevation is 30°. Find total height to 3 significant figures.
Give me a hint
Find the height above the eye level, then add 1.5.
Compare my reasoning
- Vertical difference=20 tan30°≈11.547005 m.
- Total height≈13.047005 m.
13.0 m.
Look for these in your work
- Use the horizontal distance
- Add the eye height
A 30 m vertical building is viewed by an observer with eye height 1.5 m on the same ground level. The depression angle from the top to the eyes is 40°. Find horizontal distance to 3 significant figures.
Give me a hint
The vertical difference is 28.5 m.
Compare my reasoning
- tan40°=28.5/d, so d=28.5/tan40°.
- d≈33.964977 m.
34.0 m.
Look for these in your work
- Use the eye-to-top difference
- Rearrange tangent correctly
For the illustrated 20 m horizontal distance and eye height 1.6 m, a sight line makes 55° with the vertical. Find total tower height to 3 significant figures.
Give me a hint
Convert to the horizontal elevation angle.
Compare my reasoning
- Elevation=90°−55°=35°.
- H=1.6+20 tan35°≈15.604151 m.
15.6 m.
Look for these in your work
- Identify the actual reference
- Use the complementary angle and eye height
Common mistakes
- A vertical reference needs a complementary angle.
- Do not omit eye height from total height.
What can you explain now?
Is an angle of depression measured from the vertical?
Compare with the explanation
No; it is measured downward from a horizontal.
Changing the reference from horizontal to vertical changes the angle to its complement.
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Make it stick
Is an angle of depression measured from the vertical?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.