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Heights and Distances

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high importance~2 Q in Tier 117 formulas⚡ 10 shortcuts5 subtopics
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Subtopic 1 of 5·Standard angles: 30°, 45°, 60° →

Angles of elevation and depression

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The observer's eye is the origin; the horizontal line through the eye is the reference.

  • Elevation: object is above eye level — the angle from the horizontal up to the line of sight.
  • Depression: object is below eye level — the angle from the horizontal down to the line of sight.

Height raised on a platform: always work with the eye level, not the ground. The ground distance dd and the height above eye level hh satisfy tan⁡θ=hd\tan\theta=\dfrac{h}{d}.

📖 Notes🧩 4 question types⚡ Formulas🎯 12 practice Q

Detailed notes

The picture is half the answer

Every heights-and-distances question is one right triangle wearing a story. The observer's eye is the starting point; through it runs a dashed horizontal line.

  • Object above eye level: the angle from the horizontal up to the line of sight is the angle of elevation.
  • Object below eye level: the angle down from the horizontal is the angle of depression.

The right angle sits between the horizontal and the vertical, so the trigonometry is always tan⁡θ=height above or below eye levelhorizontal distance\tan\theta = \frac{\text{height above or below eye level}}{\text{horizontal distance}}.

Eye level, not ground level

When the observer stands on a cliff, tower or building, the height asked is measured from the eye, and the ground distance runs from the base of the object. A 1.6 m tall person watching a kite: the kite's height above ground is 1.6+dtan⁡θ1.6 + d\tan\theta. Exams keep the observer's height either zero or clearly stated — but the eye is always where the horizontal starts.

Depression mirrors elevation

The angle of depression from the observer equals the angle of elevation from the object — alternate interior angles between two parallels (the two horizontals). So "from a lighthouse the depression of a boat is 30∘30^\circ" converts instantly to "from the boat the elevation of the lighthouse top is 30∘30^\circ", and the standard-angle rules apply unchanged. This one flip solves every lighthouse, cliff and aeroplane question.

Reading the question into the figure

Four phrases carry all the information:

  • "subtends an angle θ\theta at a point" → the elevation at that point is θ\theta;
  • "the angle changes from α\alpha to β\beta as he walks dd towards" → two positions on the same side, distance dd apart, β>α\beta > \alpha;
  • "observed from the top ... the angles of depression of two ships are ..." → same-side two-angle case, gap =h(cot⁡α−cot⁡β)= h(\cot\alpha - \cot\beta);
  • "shadow is kk times the height" → tan⁡(sun elevation)=1k\tan(\text{sun elevation}) = \frac{1}{k}.

Distances that are NOT on the same line

Two points not collinear with the base (e.g. two corners of a street seen from a window): each gets its own right triangle, and the two base distances may themselves form a right angle — Pythagoras joins the party. "The bearings of the two cars differ by 90∘90^\circ" means their base distances are perpendicular legs and the distance between the cars is the hypotenuse. Sketch the plan view separately from the side view; exam questions love mixing the two.

Quick revision

  • Eye = origin; horizontal = baseline; elevation up, depression down.
  • tan⁡θ=height above/below eye levelhorizontal distance\tan\theta = \frac{\text{height above/below eye level}}{\text{horizontal distance}}.
  • Depression at the top = elevation at the bottom (alternate angles).
  • Observer on a platform: add the platform height at the end.
  • "Subtends an angle at P" = elevation measured at P.

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Single angle, eye-level workvery common2 practice Q
How to spot it:

One angle and one of {height, distance} given; the other asked — including platform/cliff eye heights.

$\tan\theta = \frac{h}{d}$ w.r.t. the EYE level; add the observer's platform height if the question wants ground height.
  1. Draw the horizontal through the eye; mark the angle.
  2. Write one tangent equation in the unknown.
  3. Add the eye height (if any) before answering ground heights.

Why: everything else in the figure is decoration — one right triangle decides.

Example: A kite is flying with 30∘30^\circ elevation from a point on level ground. If the kite's thread (assumed straight) is 100100 m long, the kite's height above the ground is:

h=100sin⁡30∘=50h = 100\sin30^\circ = 50 m (the thread is the hypotenuse).

Type 2: Angle of depression conversionsvery common3 practice Q
How to spot it:

Lighthouse/cliff/aeroplane looking down at boats, cars or road junctions; depression angle stated.

Depression at the eye = elevation at the object; then $d = \frac{h}{\tan\theta}$. Two objects on one side → subtract the two horizontal distances.
  1. Flip the depression into an elevation at the object.
  2. Apply the standard-angle rule or a single tangent.
  3. For two objects, compute both distances from the base and subtract.

Why: the flip turns a "looking down" story into the ordinary up-the-tower triangle.

Example: From a 2020 m high tower the angle of depression of a car is 45∘45^\circ. The car's distance from the tower base is:

tan⁡45∘=20d⇒d=20\tan45^\circ = \frac{20}{d} \Rightarrow d = 20 m.

Type 3: Hypotenuse (thread/ladder/line-of-sight) givencommon3 practice Q
How to spot it:

The slant distance — thread, ladder, wire, 'direct distance' — is given instead of the horizontal.

$h = L\sin\theta$, $d = L\cos\theta$: the slant line is the hypotenuse, so sine and cosine (not tangent).
  1. Name the hypotenuse explicitly.
  2. Height = L·sin(angle), ground distance = L·cos(angle).
  3. Standard-angle values keep answers exact.

Why: tangent mixes the two legs; with the hypotenuse given, tan is the wrong tool.

Example: A cable of length 6060 m runs from the top of a pole to the ground making 60∘60^\circ with the ground. The pole's height is:

h=60sin⁡60∘=60×32=303h = 60\sin60^\circ = 60 \times \frac{\sqrt3}{2} = 30\sqrt3 m.

Type 4: Shadow geometrycommon3 practice Q
How to spot it:

A pole/building casts a shadow; the sun's elevation, the shadow length or the height is the unknown.

Shadow = horizontal distance: $\tan(\text{elevation}) = \frac{\text{height}}{\text{shadow}}$. Equal shadows at the same moment ⇒ equal heights; shadow $= \sqrt3 h$ means $30^\circ$, shadow $= h$ means $45^\circ$.
  1. Sun rays make the elevation angle with the ground; the triangle is pole–shadow–ray.
  2. Match the shadow:height ratio to a standard angle.
  3. Two poles at the same moment share the same sun angle — set up ratios, not two triangles.

Why: the sun's angle is the same for every object at one moment, which is the entire content of "at the same time".

Example: A vertical stick 1.51.5 m long casts a shadow 2.52.5 m long on level ground. At the same time a tower casts a shadow 7575 m long. The tower's height is:

Same sun angle: h75=1.52.5⇒h=45\frac{h}{75} = \frac{1.5}{2.5} \Rightarrow h = 45 m.

Formulas

Master relation
tan⁡θ=height above eye levelhorizontal distance\tan\theta=\frac{\text{height above eye level}}{\text{horizontal distance}}
Slant line
sin⁡θ=hL,cos⁡θ=dL(L=line-of-sight or ladder length)\sin\theta=\frac{h}{L},\quad \cos\theta=\frac{d}{L}\quad (L=\text{line-of-sight or ladder length})
Depression = elevation
angle of depression at observer=angle of elevation at object\text{angle of depression at observer}=\text{angle of elevation at object}

Shortcut tricks

⚡ Draw the horizontal through the eye

One sketch kills most errors: horizontal dashed line from the eye, the line of sight, and the right angle between them. Drop heights below the eye when the observer is elevated.

Example: From a cliff top, a boat is seen at an angle of depression of 30∘30^\circ. What angle does the line of sight make with the cliff face (vertical)?

Depression is from the horizontal, so the line of sight is 30∘30^\circ below horizontal; the vertical cliff makes 90∘−30∘=60∘90^\circ-30^\circ=60^\circ with the line of sight.

⚡ Depression at the top equals elevation at the bottom

Alternate interior angles: the depression from the observer equals the elevation measured at the object from ground level directly below — use whichever is stated.

Example: The angle of elevation of the top of a tower from a point on the ground is 45∘45^\circ. What is the angle of depression of the point from the tower top?

Equal alternate angles: 45∘45^\circ.

Where students lose marks

  • Measuring height from the ground instead of eye level when the observer is on a cliff, tower or building.

  • Using sin⁡θ\sin\theta or cos⁡θ\cos\theta where the horizontal distance makes tan⁡θ\tan\theta the right ratio.

  • Swapping elevation and depression (drawing the angle from the object instead of the horizontal at the eye).

Practice sets — 12 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 12 questions

Suggested time 7 min · wrong answers go to your mistake notebook automatically.