Three guided lessons with explanations, visual steps, practice and a final check.
Original learning companion, not full textbook exercise solutions. Covers level-ground height models, observation height, depression and a same-side two-position model. Other configurations and practical measurement uncertainty require additional lessons.
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LESSON 1
Model one angle of elevation
Choose the right triangle and a suitable trigonometric ratio.
An angle of elevation is measured upward from a horizontal line, not from the vertical.
Draw what is measured
A vertical pole stands on level ground. From a ground-level point 12 m away, its top has elevation 45°.
Label the right triangle
The horizontal leg is 12 m, the opposite vertical leg is h, and the angle at the observer is 45°.
Use tangent
tan 45° = h/12. Since tan 45° = 1, h = 12 m.
Check observation height
This model assumes observation at ground level. If an instrument is above ground, its height must be added to the calculated rise.
Worked example: At horizontal distance 10√3 m, a pole top has elevation 30°. h = 10√3 × tan 30° = 10√3/√3 = 10 m for a ground-level observer.
Watch for this: The sloping line of sight is not the horizontal distance. Decide which length the question actually gives.
Discuss: Sketch a 60° observation from 4 m away and express the pole height exactly before using a decimal approximation.
LESSON 2
Account for eye height and depression
Translate an observation into the correct vertical difference.
Two horizontal lines are parallel, so a line of sight creates equal alternate interior angles.
Separate total height and rise
An instrument stands 1.5 m above ground, 20 m horizontally from a building. Its elevation to the top is 45°.
Calculate the rise
The right triangle gives rise = 20tan45° = 20 m.
Restore the observation height
Total building height = 20 + 1.5 = 21.5 m.
Relate depression to elevation
From the top looking down along the same line of sight, the depression below its horizontal equals the elevation above the instrument horizontal.
Worked example: A lookout 18 m above a level path sees a marker at depression 45°. The corresponding elevation at the marker is 45°, so horizontal separation = 18/tan45° = 18 m.
Watch for this: An angle of depression is measured from the horizontal at the observer. It is not the angle between the sight line and a vertical wall.
Discuss: Draw two parallel horizontal lines and a sloping sight line. Mark the equal elevation and depression angles.
LESSON 3
Combine observations from two positions
Express the same height twice and solve for the unknown distance.
Moving closer to a fixed vertical tower on level ground increases its angle of elevation.
Name the nearer distance
Two ground-level observation points and the tower base lie on one straight level path. Both points are on the same side of the tower, with elevations 60° nearby and 30° from a point 20 m farther away. Let the nearer distance be x.
Write each height expression
h = x tan60° = x√3. Also h = (x + 20)tan30° = (x + 20)/√3.
Equate the same height
x√3 = (x + 20)/√3. Multiply by √3 to get 3x = x + 20; hence x = 10.
Find and verify the height
h = 10√3 m. The farther distance is 30 m, and 30/√3 = 10√3, so both observations agree.
Worked example: If the same 60°/30° observations are 12 m apart on one side, 3x = x + 12 gives x = 6 m and height 6√3 m.
Watch for this: When both observation points are on the same side, the farther distance is x plus the separation. A diagram with points on opposite sides requires a different relation.
Discuss: Keep the same angles and double the separation. Explain why both the nearer distance and tower height double.
Original learning companion, not full textbook exercise solutions. Covers level-ground height models, observation height, depression and a same-side two-position model. Other configurations and practical measurement uncertainty require additional lessons.
Original HelioLearn teaching and questions. Source-aligned companion; not an official NCERT/CBSE publication or full textbook solution bank. Independent teacher review and narrated video are pending.
