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NDA Mathematics · Formula sheet

Height & Distance formulas

16 formulas and 17 common traps for NDA Mathematics Height & Distance, grouped by subtopic.

Full notes with worked examples

Heights & Distances from Angles of Elevation

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The Right Triangle of Sight

Tangent of the angle of elevation

tan⁡θ=hd=heighthorizontal distance\tan\theta = \frac{h}{d} = \frac{\text{height}}{\text{horizontal distance}}
  • hhvertical height (opposite the angle)
  • ddhorizontal distance to the base (adjacent)
  • θ\thetaangle of elevation at the observer

A Single Observation

One triangle, three ratios

tan⁡θ=hd,sin⁡θ=hℓ,cos⁡θ=dℓ\tan\theta = \frac{h}{d}, \qquad \sin\theta = \frac{h}{\ell}, \qquad \cos\theta = \frac{d}{\ell}

Slant Distances and Half-Angle Heights

Height from slant + half-angle value

h=ℓsin⁡θ,cos⁡A2=1+cos⁡A2h = \ell\sin\theta, \qquad \cos\tfrac{A}{2} = \sqrt{\tfrac{1+\cos A}{2}}

Two Observations at Different Heights

Same base, two heights

tan⁡β=Hd,tan⁡α=H−pd\tan\beta = \frac{H}{d}, \qquad \tan\alpha = \frac{H-p}{d}

Tower Carrying a Flagstaff

Stacked heights, one base

tan⁡θ=Td,tan⁡ϕ=T+fd\tan\theta = \frac{T}{d}, \qquad \tan\phi = \frac{T+f}{d}

Angle Subtended by a Raised Segment

Subtended angle (tangent subtraction)

tan⁡α=(h2−h1) xx2+h1h2\tan\alpha = \frac{(h_2-h_1)\,x}{x^2 + h_1 h_2}

Ladders — Elevation Meets Pythagoras

Trig + length together

H=xtan⁡θ,x2+(H−k)2=L2H = x\tan\theta, \qquad x^2 + (H-k)^2 = L^2

Three Collinear Observation Points

Distance from foot at elevation θ

d=hcot⁡θ,gap=h(cot⁡θi−cot⁡θj)d = h\cot\theta, \qquad \text{gap} = h(\cot\theta_i - \cot\theta_j)

Observation Points in Different Directions

Perpendicular observers (ground Pythagoras)

z2=h2(cot⁡2y−cot⁡2x)z^2 = h^2(\cot^2 y - \cot^2 x)

A Cloud and Its Reflection in a Lake

Cloud above, image below

tan⁡α=H−pd,tan⁡β=H+pd\tan\alpha = \frac{H-p}{d}, \qquad \tan\beta = \frac{H+p}{d}

A Round Object Subtending an Angle

Subtended sphere

R=rsin⁡(α/2),h=rsin⁡βsin⁡(α/2)R = \frac{r}{\sin(\alpha/2)}, \qquad h = \frac{r\sin\beta}{\sin(\alpha/2)}

Common traps

Depression equals the elevation back

The angle of depression from the top down to a point equals the angle of elevation from that point up to the top (alternate angles between two horizontals). Drop the depression onto the ground triangle as an equal elevation — don't measure it from the vertical.

Tangent, not sine, links height to ground distance

Height vs. horizontal distance is always tan⁡θ\tan\theta. Sine and cosine bring in the slanted line of sight (hypotenuse). Reach for sin⁡θ\sin\theta only when the slant length itself is given or asked.

Keep tan⁻¹ as a ratio

When an elevation is given as tan⁡−1(5/12)\tan^{-1}(5/12), don't convert to degrees — just set tan⁡θ=5/12\tan\theta = 5/12 and substitute. Trying to find the angle numerically wastes time and invites rounding errors.

Slant uses sine, ground uses tangent

If a problem says the object is "at a distance of 1010 km from the point of observation" and gives an elevation, that distance is the SLANT line of sight — use h=ℓsin⁡θh = \ell\sin\theta. Only a stated horizontal/ground distance pairs with tan⁡θ\tan\theta.

Same base, different opposite side

Both triangles share the horizontal distance dd, but the heights opposite the angle differ: HH from the ground, H−pH-p from the raised point. Reusing HH in both equations is the standard error — subtract the observer's height.

The lower angle goes with the lower height

The smaller elevation belongs to the nearer/lower target (tower top) and the larger to the higher one (flagstaff top). Pairing tan⁡θ\tan\theta with T+fT+f by mistake flips the whole solution.

A subtended angle is a difference, not a single elevation

The angle a raised segment makes at your eye is the elevation of its TOP minus the elevation of its BOTTOM. Treat it as one elevation h/xh/x and you lose the quadratic — and the second valid position.

The ladder top is not the flagstaff top

The ladder reaches a point kk BELOW the top, so its top is at height H−kH-k, and that is what goes into Pythagoras — not HH. The elevation θ\theta, however, is measured to the flagstaff TOP at height HH. Mixing these two heights is the classic ladder slip.

Bigger angle ⇒ nearer point ⇒ smaller cotangent

Steeper elevation means you are closer to the foot, so its cotangent (distance) is smaller. Order the points by angle before subtracting distances, or the gap comes out negative.

The right angle sits at the middle observer

South of OO then east of AA makes ∠OAB=90∘\angle OAB = 90^\circ — the right angle is at AA, so OBOB (not ABAB) is the hypotenuse. Putting the right angle at OO gives the wrong Pythagoras relation.

Image depth is H + observer height, not H

The image lies HH below the lake, but the observer's eye is pp above the lake — so the image is H+pH+p below the eye, while the cloud is only H−pH-p above the eye. Forgetting to add/subtract the observer's height pp collapses the whole reflection trick.

Use half the subtended angle

The full angle α\alpha is split by the line to the centre into two equal halves; the tangent right triangle holds α/2\alpha/2, so sin⁡(α/2)=r/R\sin(\alpha/2) = r/R. Using the full α\alpha doubles your error.

Shadows, Leaning Structures & Special Geometry

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Shadows and the Sun's Elevation

Shadow of a vertical object

s=hcot⁡θ,Δs=h(cot⁡θ2−cot⁡θ1)s = h\cot\theta, \qquad \Delta s = h(\cot\theta_2 - \cot\theta_1)

Finding the New Sun Angle from a Shadow Change

New tangent, then bracket

tan⁡θ=hs1+x\tan\theta = \frac{h}{s_1 + x}

Leaning Towers — Separating Height from Lean

Two readings on a leaning tower

tan⁡α=hp−δ,tan⁡β=hq−δ\tan\alpha = \frac{h}{p-\delta}, \qquad \tan\beta = \frac{h}{q-\delta}

Chord Length of a Circle

Chord subtending central angle θ

chord=2rsin⁡θ2\text{chord} = 2r\sin\frac{\theta}{2}

Arc Length and the Equilateral-Chord Clue

Arc length

arc=rθ(θ in radians)\text{arc} = r\theta \quad (\theta \text{ in radians})

Common traps

Lower sun, longer shadow

Because s=hcot⁡θs = h\cot\theta and cotangent decreases as the angle grows, the shadow is LONGER when the elevation is SMALLER. If your lengthening s2−s1s_2 - s_1 comes out negative, you have the two angles swapped.

The new shadow is old + increase, not just the increase

The lengthening xx adds to the original shadow s1s_1; the new shadow is s1+xs_1 + x. Dividing the height by xx alone (instead of s1+xs_1 + x) over-estimates the tangent and lands you in the wrong angle band.

A leaning tower has two unknowns

Vertical height hh AND horizontal lean δ\delta are both unknown, so a single elevation reading is not enough — you must use both observation points. Treating the leaning tower like a vertical one (assuming δ=0\delta = 0) is the trap the whole set is built around.

Half the angle, not the whole angle

The chord formula uses sin⁡(θ/2)\sin(\theta/2), because the perpendicular radius bisects the central angle. Writing 2rsin⁡θ2r\sin\theta is the standard slip — and it gives a length that can wrongly exceed the diameter.

Angle must be in radians for r·θ

Arc length =rθ= r\theta only when θ\theta is in radians. Plugging in 6060 (degrees) instead of π/3\pi/3 gives an answer nearly 57×57\times too big. Convert first.

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