Core Concepts
Angle of Elevation
When you look upward from a horizontal line toward a higher object, the angle between your horizontal line of sight and your actual line of sight is the angle of elevation. The observer is below the object.
Visualise it: Stand at the base of a hill and tilt your head up to see the summit — that tilt angle from level is the angle of elevation. A crane operator looking up at the hook forms this angle.
Angle of Depression
When you look downward from a horizontal line toward a lower object, the angle between horizontal and your line of sight is the angle of depression. The observer is above the object. Note: angle of depression from A to B equals angle of elevation from B to A (alternate interior angles with a transversal cutting two horizontal parallels).
Visualise it: A lighthouse keeper looking down at a ship below — the angle their gaze drops from horizontal is the angle of depression.
Basic Right Triangle Relationship
Both elevation and depression problems reduce to a right triangle. The horizontal distance is the adjacent side; the vertical height difference is the opposite side; the direct line of sight is the hypotenuse.
Using SOH-CAH-TOA: tan(angle) = opposite / adjacent = height / horizontal distance
Two-Position Problems
When an observer moves along the ground and measures two different angles of elevation to the top of the same object, we get a system of equations. If the observer is at position A (angle α) and position B closer by distance d (angle β, where β > α):
Let h = height of object and x = horizontal distance from B. Then:
tan β = h/x → x = h/tan β, and tan α = h/(x+d) → h = d·tan α·tan β / (tan β − tan α)
Clinometer
A clinometer is an instrument used to measure angles of elevation or depression. It consists of a protractor, plumb line, and sighting tube. Surveyors use it to measure angles to hilltops, towers, and other elevated objects.
Key Formulas
Basic Height Formula
h = d × tan(α)
h = height of object above observer level, d = horizontal distance, α = angle of elevation
Height from Two Angles (same side)
h = d · tan(α) · tan(β) / (tan(β) − tan(α))
d = distance between two observation points, α = smaller angle (farther point), β = larger angle (closer point)
Width from Two Angles (opposite sides)
W = h · (cot(α) + cot(β)) = h·(1/tan α + 1/tan β)
W = width of river/valley, h = height of object on one bank
String / Hypotenuse Length
L = h / sin(α) or L = d / cos(α)
L = length of string/rope, h = vertical height, α = angle with horizontal
Elevation Calculator
Enter values below to visualise the right triangle and compute the object's total height.
Interactive Elevation Simulator
Total height = ...
Board Questions
Q1
Dhaka 2023
Angle of Elevation — Tower
4 Marks
From a point 40 m from the base of a tower, the angle of elevation of the top is 45°. Find the height of the tower.
1
Let h = height of tower, horizontal distance d = 40 m, angle α = 45°.
2
Using tan α = h / d, so tan 45° = h / 40.
3
tan 45° = 1, therefore h = 40 × 1 = 40 m.
Ans
Height of tower = 40 m ✓
Q2
Rajshahi 2024
Angle of Depression — Ship
4 Marks
From the top of a cliff 50 m high, the angle of depression of a ship at sea is 30°. Find the horizontal distance of the ship from the base of the cliff.
1
The angle of depression from cliff top to ship = 30°. The cliff height h = 50 m.
2
The angle of depression equals the angle of elevation from the ship to the cliff top (alternate angles). So tan 30° = h / d = 50 / d.
3
tan 30° = 1/√3 ≈ 0.5774, so d = 50 / tan 30° = 50√3.
4
d = 50 × 1.732 ≈ 86.6 m.
Ans
Distance of ship from cliff base = 50√3 ≈ 86.6 m ✓
Q3
Chittagong 2022
Observer Height — Tree
5 Marks
A man 1.6 m tall stands 20 m from a tree. He looks up at 37° to see the top of the tree. Find the height of the tree. (Use tan 37° ≈ 0.75)
1
The man's eye level is approximately at his height = 1.6 m. Horizontal distance d = 20 m, angle α = 37°.
2
Height above eye level: h' = d × tan 37° = 20 × 0.75 = 15 m.
3
Total tree height = height above eye level + observer's eye level height: H = 15 + 1.6 = 16.6 m.
Ans
Height of tree = 16.6 m ✓
Q4
Jessore 2025
Two-Position Formula
6 Marks
From two points A and B, 100 m apart on the same side of a tower, the angles of elevation of the top of the tower are 30° and 60° respectively (B is closer). Find the height of the tower.
1
Let h = height of tower, x = distance from B to tower base. d = AB = 100 m. Angles: α = 30° (from A), β = 60° (from B).
2
From B: tan 60° = h/x → x = h/tan 60° = h/√3.
3
From A: tan 30° = h/(x+100) → x + 100 = h/tan 30° = h√3.
4
Subtracting: 100 = h√3 − h/√3 = h(√3 − 1/√3) = h·(3−1)/√3 = 2h/√3.
5
Solve: h = 100√3/2 = 50√3 ≈ 86.6 m.
Ans
Height of tower = 50√3 ≈ 86.6 m ✓
Q5
Comilla 2021
Flagpole on Building
6 Marks
A flagpole stands on top of a building 10 m high. From ground level 25 m away, the angle of elevation to the top of the flag is 60° and to the top of the building is 45°. Find the height of the flagpole.
1
Let P = height of flagpole. Building height = 10 m. Horizontal distance d = 25 m.
2
Verify building: tan 45° = 10/25 → 1 = 10/25 = 0.4. This doesn't hold exactly — in fact the problem provides the angle to building top as 45°, so we use the actual building height from angle: h_bldg = 25 × tan 45° = 25 m (this supersedes the "10m" given — alternatively use the stated height). Using stated angle: height to flag top = 25 × tan 60° = 25√3 m.
3
Height to building top = 25 × tan 45° = 25 × 1 = 25 m.
4
Flagpole height = 25√3 − 25 = 25(√3 − 1) ≈ 25 × 0.732 ≈ 18.3 m.
Ans
Height of flagpole = 25(√3 − 1) ≈ 18.3 m ✓
Q6
Barisal 2024
Depression — Distance Between Cars
5 Marks
A bird on top of a tree sees two cars directly ahead. The angles of depression are 45° and 30°. The tree is 60 m high. Find the distance between the two cars.
1
Tree height h = 60 m. Let car C₁ be closer (depression 45°) and car C₂ be farther (depression 30°).
2
Distance to C₁: d₁ = h / tan 45° = 60 / 1 = 60 m.
3
Distance to C₂: d₂ = h / tan 30° = 60 / (1/√3) = 60√3 ≈ 103.9 m.
4
Distance between cars: D = d₂ − d₁ = 60√3 − 60 = 60(√3 − 1) ≈ 43.9 m.
Ans
Distance between cars = 60(√3 − 1) ≈ 43.9 m ✓
Q7
Sylhet 2023
Opposite Banks — River Width
6 Marks
From opposite banks of a river, the angles of elevation of the top of a hill are 30° and 45°. The hill is 100 m high. Find the width of the river.
1
Let the hill sit at the edge of one bank. W = width of river. Height h = 100 m.
2
From the same bank as the hill (angle 45°): observer is at the base of the hill, so the distance is 0 — this angle must be from the opposite side. Let x₁ = distance from hill base to near bank, x₂ = width of river.
3
More precisely: let d₁ = distance from bank-1 (angle 30°) and d₂ = distance from bank-2 (angle 45°). These are the two horizontal distances: d₁ = h/tan 30° = 100√3 m and d₂ = h/tan 45° = 100 m.
4
If the hill stands on the bank, then the width W equals d from the opposite bank. The hill stands between the two observers — the sum of the two distances equals the width only if the hill is at an edge. Using the standard interpretation: hill top is at one bank, observers on opposite banks. Width = d₂ = 100 m. The other bank gives angle 30°, so that distance = 100√3 from that bank. Width = 100√3 + 100 = 100(√3+1) ≈ 273.2 m if the hill is in the middle of neither bank but both observers are at water's edge and hill is at edge of one bank the width equals d from the far bank = 100√3 ≈ 173.2 m.
5
Standard solution: hill at one bank, opposite bank observer sees 30°. Width = h/tan 30° = 100√3 ≈ 173.2 m.
Ans
Width of river = 100√3 ≈ 173.2 m ✓
Q8
Dinajpur 2022
Kite String Length
4 Marks
A kite is flying at a height of 80 m. The string makes an angle of 60° with the ground. Find the length of the string (assuming the string is straight).
1
Height h = 80 m, angle with ground α = 60°. The string is the hypotenuse L.
2
Using sine: sin 60° = h / L → L = h / sin 60°.
3
sin 60° = √3/2, so L = 80 / (√3/2) = 160/√3 = 160√3/3 ≈ 92.4 m.
Ans
Length of string = 160/√3 = 160√3/3 ≈ 92.4 m ✓
Formula Sheet
Height from single angle
h = d · tan α
String / hypotenuse
L = h / sin α
Two-position (same side)
h = d·tan α·tan β / (tan β − tan α)
Horizontal distance
d = h / tan α = h · cot α
tan values
tan30°=1/√3 tan45°=1 tan60°=√3
sin values
sin30°=1/2 sin45°=1/√2 sin60°=√3/2