Measured from the horizontal: An angle of elevation is measured upward from the horizontal to a point above you; an angle of depression is measured downward from the horizontal to a point below you.
Interactive: switch between an angle of elevation (looking up) and an angle of depression (looking down) — both measured from the horizontal.
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Always sketch the horizontal: Draw the horizontal line at the observer's eye first — the angle is between that line and the line of sight.
Not from the vertical: These angles are from the horizontal, never the vertical — a common slip that flips the right ratio.
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It's a right-angled triangle: Elevation problems form a right-angled triangle: the height is opposite the angle, the ground distance is adjacent. Use tan (or sin/cos as needed).
IB-style question — height of a tower
From a point 50 m from the base of a tower, the angle of elevation to the top is 30°. Find the height of the tower.
Step by step
- Opposite (height) and adjacent (50) → tan.
- Solve.
Final answer
About 28.9 m.
The elevation triangle: the 30° angle of elevation is at the observer, the tower height h is opposite it, and the 50 m ground distance is adjacent. tan 30° = h ÷ 50 → h ≈ 28.9 m.
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Add eye height if given: If the angle is measured from a person's eye (e.g. 1.6 m up), add that to the triangle's height for the true total.
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Depression down = elevation up: The angle of depression from a high point down to an object equals the angle of elevation from the object back up — they're alternate angles between two parallel horizontals.
Tap 'Depression' (or 'Both'): the angle of depression from the high point and the angle of elevation back up from the object are equal — alternate angles between the two parallel horizontals. That lets you mark the angle low down, inside the right-angled triangle, as in the boat question below.
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IB-style question — depression to a boat
From the top of a 40 m cliff, the angle of depression to a boat is 20°. Find the boat's distance from the base of the cliff.
Step by step
- The depression (20°) equals the elevation from the boat, so tan 20° = 40/d.
- Solve.
Final answer
About 110 m.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Use the angle inside the triangle: Mark the equal alternate angle at the bottom so the right-angled triangle has the angle in a usable place.
Non-right triangle → sine/cosine rule: When two observation points (or two towers) and an object form a non-right triangle, the sine or cosine rule does the work — exactly as in the audited two-tower elevation question.
IB-style question — two towers
From a point between two tower tops, the slant distances to the tops are 100 m and 160 m, and the distance between the two tops is 170 m. Find the angle at the point.
Step by step
- Three sides → cosine rule for the angle.
- Evaluate.
Final answer
About 78.0°.
The two towers and the point form a triangle: the slant distances 100 and 160 meet at the point (the angle θ), and 170 is the side opposite. Three sides → cosine rule for θ: cos θ = (100² + 160² − 170²)/(2·100·160) → θ ≈ 78.0°.
Interactive diagram
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GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Elevation gives one angle: An elevation angle from the same point becomes one angle of the bigger triangle — then a right-triangle step finds a height.