Angles of Elevation and Depression

How tall is that radio mast on the hill? How far out to sea is that ship? How high is a passing plane? You can't lower a tape measure from a cloud, and you certainly can't walk to a ship. Yet with nothing more than a measured angle and a little trigonometry, all three become easy — you can measure things you could never reach.

The trick is the line of sight: the straight line from your eye to the object. Tilt your gaze up to the top of a tower and the angle your line of sight makes with the horizontal is the angle of elevation. Stand on a cliff and look down at a boat, and the angle below the horizontal is the angle of depression. Feed that one angle into a right-angled triangle, and a real height or distance drops out.

The two angles

The angle of elevation is measured up from the horizontal — the angle you tilt your gaze to look up at a treetop or the top of a building.
The angle of depression is measured down from the horizontal — the angle you tilt your gaze to look down from a cliff or a window at something below.

Both are measured from the horizontal — the flat, eye-level line — never from the vertical. And here is a fact worth banking: when two people look at each other — you up at the cliff-top, someone down at the boat — the angle of elevation from below equals the angle of depression from above. The two horizontals are parallel, so these are alternate angles between parallel lines.

To solve a problem, draw a right-angled triangle: the horizontal distance, the vertical height, and the slanted line of sight. Then it's just SOH-CAH-TOA again.

For an observer looking at an object across a horizontal:

the angle of elevation is the angle measured up from the horizontal;
the angle of depression is the angle measured down from the horizontal;
between the same two points the two angles are equal (alternate angles between parallel horizontals);
model the situation with a right-angled triangle and solve with \sin, \cos or \tan.

Worked example 1 — how tall is the tower?

A surveyor stands 50\text{ m} from the foot of a tower. Looking up at the top, she measures an angle of elevation of 38^\circ. How tall is the tower?

Draw the right triangle: the horizontal distance d = 50 is adjacent to the angle, and the height h is opposite it. Opposite and adjacent means TOA:

\tan 38^\circ = \frac{h}{50} \quad\Longrightarrow\quad h = 50\,\tan 38^\circ \approx 50 \times 0.7813 = 39.1\text{ m}.

The tower is about 39 m tall — measured without ever leaving the ground.

Worked example 2 — how far away is the boat?

From the top of an 80\text{ m} cliff, the angle of depression of a boat is 25^\circ. How far is the boat from the base of the cliff?

The angle of depression from the cliff-top equals the angle of elevation from the boat (alternate angles), so inside the right triangle the angle at the boat is 25^\circ. The cliff height h = 80 is opposite that angle, and the distance d is adjacent — TOA again:

\tan 25^\circ = \frac{80}{d} \quad\Longrightarrow\quad d = \frac{80}{\tan 25^\circ} \approx \frac{80}{0.4663} = 171.6\text{ m}.

The boat is about 172 m out — found from a single angle at the cliff-top.

Worked example 3 — a two-stage problem

Sometimes one angle isn't enough, so we take two. Standing on level ground, you measure the angle of elevation to the top of a tower as 30^\circ. You walk 40\text{ m} straight towards it and measure again: 45^\circ. How tall is the tower — and you never measured your distance to it?

Let the tower have height h and let your closer position be x metres from its base. From the near point, \tan 45^\circ = h/x, and since \tan 45^\circ = 1, that gives h = x. From the far point (a further 40\text{ m} back),

\tan 30^\circ = \frac{h}{x + 40}.

Substitute x = h and use \tan 30^\circ = 1/\sqrt{3}:

\frac{1}{\sqrt 3} = \frac{h}{h + 40} \;\Longrightarrow\; \sqrt 3\,h = h + 40 \;\Longrightarrow\; h = \frac{40}{\sqrt 3 - 1} \approx 54.6\text{ m}.

The tower is about 55 m tall — measured with two angles and one pace count.

A worked picture

Step through it: place the observer and the object, mark the angle of elevation, then read off the ratio that links the height to the distance.

These three trip almost everyone up:

Always from the horizontal. If you measure the angle down from the vertical instead — say 55^\circ from straight up — you get the complement, and the elevation is really 90^\circ - 55^\circ = 35^\circ. Feed the wrong angle into \tan and your answer is nonsense. Draw the horizontal line first, every time.
Depression equals elevation. The angle of depression from A down to B equals the angle of elevation from B up to A — they are alternate angles between two parallel horizontals. It is easy to draw the depression angle inside the triangle by mistake; instead, carry it across to the object as the equal elevation angle.
Match the ratio to the sides. Opposite and adjacent → tan; opposite and hypotenuse (the line of sight) → sin. Label the triangle before you reach for a button.

This is the everyday superpower of elevation and depression: they let you measure the unreachable. Point a clinometer — a little protractor with a weighted string or a sighting tube — at a mountain peak, read the angle, pace out a known distance, and trig hands you the height. The same idea gauges the depth of a canyon, the altitude of an aircraft, and the width of a river you never cross.

Sailors used exactly this for centuries, taking the angle to a lighthouse or a distant ship to judge how far off it was before radar existed. Ancient astronomers pushed the idea to the sky, using elevation angles to estimate the height of the atmosphere and even rough distances to the Moon. One angle, one baseline, and suddenly the whole world is within reach of your protractor.

The whole game is matching the ratio to the two sides you have. Label the triangle from the angle you know: the side facing it is the opposite, the side along it (not the hypotenuse) is the adjacent, and the slanted line of sight is the hypotenuse. Then SOH-CAH-TOA picks the button for you:

have opposite and adjacent (a height and a ground distance)? → \tan;
have opposite and hypotenuse (a height and a line-of-sight length)? → \sin;
have adjacent and hypotenuse? → \cos.

Most elevation and depression problems hand you a height and a horizontal distance, so \tan does most of the heavy lifting — but a taut rope or a line-of-sight length is your cue to reach for \sin.