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Heights & DistancesFoundation

Measure a tower without touching it — interactive Mathematics simulation for IIT-JEE.

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Concept

Surveying with a protractor: measure the angle of elevation of the top from two points in line with the base, a known distance d apart. Two right triangles share the same height — eliminate the unknown base distance and the height falls out: h=dcot⁡α−cot⁡βh = \frac{d}{\cot\alpha - \cot\beta}.

Key formula

h=dcot⁡α−cot⁡β=dtan⁡αtan⁡βtan⁡β−tan⁡αh = \frac{d}{\cot\alpha - \cot\beta} = \frac{d\tan\alpha\tan\beta}{\tan\beta - \tan\alpha}

Derivation

Let x = distance from the NEAR point to the base. Then x=hcot⁡βx = h\cot\beta and x+d=hcot⁡αx + d = h\cot\alpha.

Subtract: d=h(cot⁡α−cot⁡β)d = h(\cot\alpha - \cot\beta). Special exam case: β = 2α makes the far triangle isosceles — the slant from the far point equals... use sine rule shortcuts. The 30°/60° pair gives h = d·√3/2·... just plug in: h = d/(√3 − 1/√3) = d√3/2.

Scenarios to explore

  • Heights & Distances — Measure a tower without touching it — angles of elevation.

Real-world applications

  • Classic surveying, mountain heights (how Everest was first measured).
  • Angle of depression problems (same triangles, viewed from the top).
  • Lighthouse/ship, balloon/river-bank variants.

JEE exam tips

  • cot forms are cleaner than tan for two-observation problems.
  • 45° observations mean distance = height — free simplifications.
  • String/kite problems: the string is the HYPOTENUSE, h = L·sinθ.

Common mistakes

  • Mixing up which angle pairs with which distance (FAR point ↔ SMALLER angle).
  • Elevation measured from eye level — add observer height if given.
  • Depression angle taken from the vertical instead of horizontal.

Exam traps to avoid

  • Angle of depression from the top = angle of elevation from the bottom (alternate angles).
  • If the observer's height matters, the 'tower' in equations is (H − observer height).