Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Identify opposite and adjacent sides relative to an angle.
- Use tangent to find a vertical rise.
- Add the observer's eye height and report a sensible approximation.
Before you start
Right triangles, multiplication, and using a calculator in degree mode.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
An observer stands 20 m horizontally from a tree on level ground. The angle from eye-level horizontal to the top is 35°, and the observer's eyes are 1.6 m above the ground. Estimate the tree's height.
Why this math matters
A right triangle connects a distance you can measure to a height you want to estimate. This is the core mathematical idea behind indirect height measurement. Drawing the triangle at eye level prevents a common error: forgetting that the angle starts above the ground.
Set up the model
A useful answer starts with clear assumptions:
- The ground between observer and tree is level, and the tree is vertical.
- The 20 m is horizontal distance to the base, and 35° is measured from horizontal at eye level.
- Measurements are approximate. This is an educational estimate, not a survey or a safety assessment.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Estimate a tree's height with trigonometry
PausedQuestion: Start with the question. Paused.
Question
Start with the question
An observer stands 20 m horizontally from a tree on level ground. The angle from eye-level horizontal to the top is 35°, and the observer's eyes are 1.6 m above the ground. Estimate the tree's height.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Name the unknown rise
h = rise from the observer's eye level to the tree top
The right triangle's horizontal leg is 20 m. Its vertical leg is h, which does not include the 1.6 m from the ground to eye level.
Choose the tangent ratio
tan(35°) = h / 20
The unknown vertical leg is opposite the 35° angle and the known horizontal leg is adjacent. Tangent relates exactly those two sides.
Solve for rise above eye level
h = 20 tan(35°) ≈ 14.004 m
In degree mode, tan(35°) is approximately 0.700208. Keep calculator precision until the total height is found.
Include eye height
H = h + 1.6 ≈ 15.604 m ≈ 15.6 m
The total height combines the rise above eye level and the distance from eye level to the level ground. The final decimal is an approximation, not a claim of survey accuracy.
The result
Under the stated assumptions, the tree is approximately 15.6 m tall.
If the ground slopes or the tree leans, this single right-triangle model may no longer represent the geometry. Also, an angle measured a little too high produces a larger estimate. More decimal places cannot repair uncertain input measurements.
Common mistakes to catch
- Using sine would require the sloping line-of-sight distance, which is not given.
- Forgetting eye height underestimates the total height by 1.6 m in this model.
- Radians mode gives the wrong result when the stated angle is in degrees.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
On level ground, an observer is 12 m horizontally from a vertical pole. The angle of elevation is 45° and eye height is 1.5 m. Estimate the total pole height.
Show a hint
tan(45°) = 1. Find the rise above eye level, then add 1.5 m.
Reveal answer and explanation
13.5 m
Rise above eye level = 12 × 1 = 12 m. Total height = 12 + 1.5 = 13.5 m under the model's assumptions.
Practice 2
A vertical model tower is 11.5 m tall. An observer's eye height is 1.5 m and the angle of elevation is 30°. On level ground, what horizontal distance does the model imply?
Show a hint
Subtract eye height first. Then use tan(30°) = rise/distance.
Reveal answer and explanation
Approximately 17.3 m
The rise is 11.5 − 1.5 = 10 m. The distance is 10/tan(30°) = 10√3 m, approximately 17.3 m.
Take the idea with you
For another indirect-measurement problem, draw the observer's horizontal line first. Label what is measured, what is assumed, and which length the unknown actually represents.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Use an identity without losing the quadrant
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