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CDS Mathematics · Heights and Distances

Two Points of Observation

Two observations of one height give two ground distances, h cot α and h cot β; the question hands you their difference or their sum.

All Heights and Distances notesCDS Mathematics strategy

Why this matters

Nineteen PYQs, four of them HARD — the biggest page of the chapter. Moving towards a tower, a car approaching, two stones on a road: the gap between the points is h(cot β − cot α). Opposite sides: the distances add. Complementary angles give the shortcut h² = pq.

Concept 1 of 2: Difference and sum of the distances

Each observation of the same height fixes one ground distance, hcot⁡θh\cot\theta. If both points are on the same side, walking between them covers the difference; if the object is between them, their gap is the sum.

Definition

  • Same side, angles α<β\alpha < \beta: gap =h(cot⁡α−cot⁡β)= h(\cot\alpha - \cot\beta).
  • Opposite sides: gap =h(cot⁡α+cot⁡β)= h(\cot\alpha + \cot\beta).
  • 30∘30^\circ and 60∘60^\circ: cot⁡30∘−cot⁡60∘=23\cot 30^\circ - \cot 60^\circ = \dfrac2{\sqrt3}; cot⁡30∘+cot⁡60∘=43\cot 30^\circ + \cot 60^\circ = \dfrac4{\sqrt3}.
  • Uniform speed: times are in the ratio of the distances covered.
  • A flagstaff on a tower: two heights, the same ground distances.

Same side

d=h(cot⁡α−cot⁡β)d = h(\cot\alpha - \cot\beta)

Worked example

From two points 2020 m apart on the same side of a tower, the elevations of its top are 30∘30^\circ and 60∘60^\circ. Find its height.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2026 · CDS (I) 2026 — Elementary Mathematics · Q45Moderate

Example 1 · Heights and Distances · Two Points of Observation

Two persons are on diametrically opposite sides of a tower. They measure the angles of elevation of the top of the tower as 30∘30^\circ and 60∘60^\circ respectively. If the height of the tower is 100 m, what is the approximate distance between the two persons ?

Same side or opposite sides?

'On either side of the tower' or 'on the opposite banks' means the distances ADD. Subtracting them gives an option that is always there.

Nearer point, larger angle

The point closer to the tower sees the larger elevation. Swapping the angles makes the gap negative.
Drill 13 more on difference and sum of the distances

Concept 2 of 2: Complementary angles

If the two elevations add to 90∘90^\circ, then tan⁡α⋅tan⁡β=1\tan\alpha \cdot \tan\beta = 1. Multiplying h=ptan⁡αh = p\tan\alpha by h=qtan⁡βh = q\tan\beta removes the angles entirely.

Definition

  • Elevations α\alpha and 90∘−α90^\circ - \alpha from distances pp and qq: h2=pqh^2 = pq.
  • With heights h1h_1 and h2h_2 seen from one point: h1h2=xyh_1h_2 = xy.
  • Conversely, tan⁡αtan⁡β=1\tan\alpha\tan\beta = 1 means α+β=90∘\alpha + \beta = 90^\circ.
  • A double angle, β=2α\beta = 2\alpha: use tan⁡2α=2tan⁡α1−tan⁡2α\tan 2\alpha = \dfrac{2\tan\alpha}{1 - \tan^2\alpha}.

Complementary elevations

h2=pqh^2 = pq

Worked example

The elevations of a tower's top from points 44 m and 1616 m from its foot (same line) are complementary. Find its height.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2020 · CDS (I) 2020 — Elementary Mathematics · Q57Moderate

Example 2 · Heights and Distances · Two Points of Observation

The angles of elevation of the top of a tower from two points at distances pp and qq from the base and on the same straight line are 27∘27^\circ and 63∘63^\circ respectively. What is the height of the tower?

Heights multiply, distances multiply

h2=pqh^2 = pq uses the PRODUCT of the distances. With 44 and 1616, the height is 88, not 1010, their average.
Drill 4 more on complementary angles

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Watch out for (3)

Test yourself on Heights and Distances

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Heights and Distances chapter testDrill all 19 questions