MISSION · Geometry
Angles of Elevation and Depression
Measuring things you cannot reach, with one angle and one distance
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The idea in one sentence
An angle of elevation or depression is always measured from the horizontal, which turns any “how tall is it” question into a right triangle you can solve with trig ratios.
The two angles
Stand on the ground and look up at the top of a tower. The angle of elevation is the angle between your horizontal line of sight and the line to the top.
Stand on a cliff and look down at a boat. The angle of depression is the angle between your horizontal and the line down to the boat.
Both are measured from the horizontal, never from the vertical. Measuring from a wall or a flagpole instead gives the complement of the right angle, and an answer that is wrong by exactly \(90° - \theta\).
They are equal, and that is useful
The angle of depression from the cliff to the boat equals the angle of elevation from the boat to the cliff.
The two horizontals — one at the cliff top, one at sea level — are parallel, and the line of sight is a transversal, so the two angles are alternate angles and therefore equal.
This matters because questions are usually posed from the high point while the triangle is easier to label from the low one. Moving the angle down to the ground is legitimate, and it puts the angle inside the triangle where you can use it.
Solving one
A tower is 50 m away, with an angle of elevation of 30°. How tall is it?
Sketch the right triangle. The 50 m is the horizontal — adjacent to the angle. The height is opposite. Opposite and adjacent means tangent:
\[\tan 30° = \frac{h}{50} \quad \Rightarrow \quad h = 50\tan 30° \approx 28.9 \text{ m}\]Three steps, always the same. Sketch and label which side is opposite, adjacent and hypotenuse relative to the angle you know. Pick the ratio containing the side you want and the side you have. Solve.
If the question asks for an angle instead, you have two sides and need the inverse: \(\theta = \tan^{-1}\!\left(\frac{h}{d}\right)\).
The eye-height trap
If a 1.7 m tall surveyor measures an elevation of 30°, the 28.9 m is the height above the instrument, not above the ground. The tower is 30.6 m tall.
Exam questions include eye height precisely to see whether you noticed. Whenever a person’s height is mentioned at all, it is there to be added at the end.
Two-triangle problems
Harder questions give two angles from two positions and no distance at all.
From one point the elevation of a tower is 30°; 40 m closer it is 45°. Find the height.
Let the height be \(h\) and the nearer distance be \(x\). Then \(\tan 45° = \frac{h}{x}\), so \(x = h\). And \(\tan 30° = \frac{h}{x + 40}\), so \(h = (h + 40)\tan 30°\). Solving,
\[h(1 - \tan 30°) = 40 \tan 30° \quad \Rightarrow \quad h = \frac{40 \tan 30°}{1 - \tan 30°} \approx 54.6 \text{ m}\]Two triangles sharing the height, two equations, one unknown eliminated. That structure covers nearly every “two positions” problem, and it is the reason 45° is such a common choice in these questions — it makes one distance equal to the height.
Common mix-ups
- Measuring from the vertical. Both angles come off the horizontal.
- Using sine when you have opposite and adjacent. Label the triangle before choosing.
- Forgetting eye height when the question supplies it.
- Putting the angle of depression inside the triangle without moving it to the far corner first.
- Calculator in the wrong mode. Degrees, unless the question uses radians.
Check yourself
Find the height of a tower 50 m away at an elevation of 30°, say whether the angles of elevation and depression between two points are equal, and name the ratio linking opposite to adjacent.
\(50\tan 30° \approx 28.9\) m, so 29 m to the nearest metre. The two angles are equal, being alternate angles between parallel horizontals. And opposite over adjacent is the tangent.
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