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SPI Mathematics · Formula sheet

Heights and Distances formulas

7 formulas and 17 common traps for SPI Mathematics Heights and Distances, grouped by subtopic.

Full notes with worked examples

Angles of Elevation and Depression

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Line of sight, elevation and depression

Depression equals elevation

angle of depression from the top=angle of elevation from the ground point\text{angle of depression from the top} = \text{angle of elevation from the ground point}

Height and distance through tan

Height and ground distance

tan⁡θ=hd,h=dtan⁡θ,d=htan⁡θ\tan\theta = \frac{h}{d}, \qquad h = d\tan\theta, \qquad d = \frac{h}{\tan\theta}
  • hhthe upright height
  • ddthe distance along the ground to its foot

The sun's elevation and a shadow

Sun's elevation from a shadow

tan⁡θ=heightshadow\tan\theta = \frac{\text{height}}{\text{shadow}}

Common traps

Measuring depression from the tower

The angle of depression is measured down from the horizontal at the eye. The angle between the line of sight and the upright tower is 90∘90^\circ minus it, so using that angle swaps tan for cot.

Putting the depression angle at the top of the triangle

Inside the triangle, the angle at the top is not the angle of depression. Move the angle to the ground point, where it becomes the angle of elevation, and work from there.

Turning tan upside down

tan⁡θ\tan\theta is height over distance. Writing distance over height gives cot⁡θ\cot\theta, and with it the other standard angle: 60∘60^\circ where the answer is 30∘30^\circ.

Dividing when you should multiply

To find a height, multiply the distance by tan⁡θ\tan\theta. To find a distance, divide the height by it. A quick check: at 45∘45^\circ the two are equal, and above 45∘45^\circ the height is the longer one.

Reading the ratio the wrong way

Height to shadow 1:31 : \sqrt{3} means the shadow is longer, so the sun is low: 30∘30^\circ. Reading it as shadow to height gives 60∘60^\circ, which is printed as a wrong option.

A high sun does not lengthen the shadow

As the sun's elevation grows from 30∘30^\circ to 60∘60^\circ, the shadow of the same pole shrinks to a third of its length. A bigger angle always means a shorter shadow.

Kites, Ropes and Slopes

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The slanting side is the hypotenuse

Height and ground distance from a slant length

h=Lsin⁡θ,d=Lcos⁡θ,L=hsin⁡θh = L\sin\theta, \qquad d = L\cos\theta, \qquad L = \frac{h}{\sin\theta}
  • LLthe slant length (string, rope, slide, ramp)
  • hhthe height of its top end
  • ddthe ground distance it covers

Common traps

Using tan with a slant length

tan joins the height and the ground distance only. When the string or rope length is given or asked, the hypotenuse is involved, so use sin or cos.

Swapping sin and cos

The angle sits at the ground. The height is across from it (sin) and the ground distance touches it (cos). A 30 m ladder at 30∘30^\circ reaches 15 m up the wall, and its foot is 15315\sqrt{3} m out.

Using the taller pole's full height

A wire between poles of 20 m and 12 m rises only 8 m. The ground below the shorter pole's top is not part of the triangle.

Calling the broken part the whole tree

The broken part is only the hypotenuse. If the question asks for the tree's height before it broke, add the standing part to the broken part.

Two Angles of Elevation

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Complementary angles from two points: h = √(ab)

Height from complementary angles

h=abh = \sqrt{ab}
  • a,ba, bdistances of the two points from the foot of the tower
  • hhheight of the tower

An observer who walks, or a shadow that changes

Height from two angles and the walk between them

h=dcot⁡α−cot⁡βh = \frac{d}{\cot\alpha - \cot\beta}
  • dddistance walked towards the foot (or change in shadow length)
  • ααthe smaller angle, from the farther point
  • ββthe larger angle, from the nearer point

A flagstaff on a tower, a cloud and its reflection

Reflection in still water

rise to the cloud=C−h,drop to its image=C+h\text{rise to the cloud} = C - h, \qquad \text{drop to its image} = C + h

Common traps

Adding or multiplying the distances

The height is the square root of the product of the two distances, not their sum and not their product. With distances 4 m and 9 m it is 6 m, not 13 m or 36 m.

Giving the farther point the bigger angle

The closer you stand to a tower, the steeper you look up. The point at the smaller distance sees the larger angle, so match 60∘60^\circ with the nearer point.

Using the walk as a whole distance

The distance walked is the gap between the two points, not either point's distance from the foot. Give each point its own distance, x and x+dx + d.

Subtracting tan instead of cot

The ground distances are hcot⁡αh\cot\alpha and hcot⁡βh\cot\beta, so the formula has cot⁡α−cot⁡β\cot\alpha - \cot\beta below the line. A denominator of tan⁡β−tan⁡α\tan\beta - \tan\alpha is a printed wrong option.

Leaving a speed in km per second

2 km in 10 seconds is 2×360010=7202 \times \frac{3600}{10} = 720 km/h. Multiply by 3600 and divide by the seconds; dividing by 3600 gives a speed far too small.

Using the flagstaff alone for its top

The top of a flagstaff on a tower is the tower plus the flagstaff above the ground. Putting only the flagstaff's length over the distance gives an angle that is far too small.

Measuring the reflection from the observer

The image is as far below the water as the cloud is above it. From a point above the lake, the drop to the image is the cloud's height plus the observer's height, not minus.

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