SPI Mathematics · Heights and Distances

Angles of Elevation and Depression

An angle of elevation is how far you look up from the horizontal to see an object, an angle of depression is how far you look down, and tan links the height to the distance along the ground.

All Heights and Distances notes

Why this matters

Eleven questions in the bank, all EASY. Ten are one right triangle with an upright height and flat ground, solved with tan: find a height, a distance or the angle. Five of those are a pole and its shadow under the sun, and the eleventh asks for the instrument that measures these angles.

Concept 1 of 3: Line of sight, elevation and depression

The line of sight is the straight line from your eye to the object. If the object is above you, the angle between that line and the horizontal is the angle of elevation. If it is below you, as when you look down from a tower, the angle is the angle of depression.

Definition

Both angles are measured from the horizontal at the eye, never from the vertical.
The angle of depression from the top equals the angle of elevation from the ground point. The horizontal at the top and the ground are parallel, so these are alternate angles.
So a depression question becomes the usual triangle: put the same angle at the ground point.
A clinometer (inclinometer) is the instrument that measures these angles.

Depression equals elevation

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

Worked example

From the top of a cliff 60 m high, the angle of depression of a boat is

30^\circ

. How far is the boat from the foot of the cliff?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Heights and Distances · Angles of Elevation and Depression

A player sitting on the top of a tower of height 40 m observes the angle of depression of a ball lying on the ground is

60^\circ

. The distance between the foot of the tower and the ball is

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^\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.

Drill 2 more on line of sight, elevation and depression

Concept 2 of 3: Height and distance through tan

The height stands upright on flat ground, so the height, the ground distance and the line of sight make a right triangle. The height is opposite the angle and the ground distance is next to it, so tan joins them.

Definition

$\tan\theta = \dfrac{\text{height}}{\text{distance}}$ .
Height $=$ distance $\times \tan\theta$ ; distance $=$ height $\div \tan\theta$ .
Given both lengths, find $\tan\theta$ and read the angle from the table: $\frac{1}{\sqrt{3}}$ is $30^\circ$ , 1 is $45^\circ$ , $\sqrt{3}$ is $60^\circ$ .
The distance is measured along the ground to the foot of the height.

Height and ground distance

\tan\theta = \frac{h}{d}, \qquad h = d\tan\theta, \qquad d = \frac{h}{\tan\theta}

hthe upright height
dthe distance along the ground to its foot

Worked example

The top of a tower is seen at an angle of elevation of

60^\circ

from a point 25 m from its foot. Find the height of the tower.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Heights and Distances · Angles of Elevation and Depression

The angle of elevation of the top of a 15 m high tower at a point

15\sqrt{3}

m away from the base of the tower is:

Turning tan upside down

\tan\theta

is height over distance. Writing distance over height gives

\cot\theta

, and with it the other standard angle:

60^\circ

where the answer is

30^\circ

Dividing when you should multiply

To find a height, multiply the distance by

\tan\theta

. To find a distance, divide the height by it. A quick check: at

45^\circ

the two are equal, and above

45^\circ

the height is the longer one.

Drill 2 more on height and distance through tan

Concept 3 of 3: The sun's elevation and a shadow

Sunlight comes in straight lines. The pole, its shadow on the ground and the ray from the top of the pole to the tip of the shadow make a right triangle, and the angle at the tip of the shadow is the sun's elevation (its altitude).

Definition

$\tan(\text{sun's elevation}) = \dfrac{\text{height of the object}}{\text{length of its shadow}}$ .
A ratio of height to shadow of $1 : \sqrt{3}$ gives $30^\circ$ ; $1 : 1$ gives $45^\circ$ ; $\sqrt{3} : 1$ gives $60^\circ$ .
The higher the sun, the shorter the shadow.

Sun's elevation from a shadow

\tan\theta = \frac{\text{height}}{\text{shadow}}

Worked example

A tower 21 m high casts a shadow

21\sqrt{3}

m long. Find the sun's elevation.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Heights and Distances · Angles of Elevation and Depression

A pole casts a shadow of length

2\sqrt{3}

m on the ground when the sun's elevation is

60^\circ

. The height of the pole is

Reading the ratio the wrong way

Height to shadow

1 : \sqrt{3}

means the shadow is longer, so the sun is low:

30^\circ

. Reading it as shadow to height gives

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^\circ

60^\circ

, the shadow of the same pole shrinks to a third of its length. A bigger angle always means a shorter shadow.

Drill 4 more on the sun's elevation and a shadow

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 (3)

Line of sight, elevation and depression
Depression equals elevation
$\text{angle of depression from the top} = \text{angle of elevation from the ground point}$
Height and distance through tan
Height and ground distance
$\tan\theta = \frac{h}{d}, \qquad h = d\tan\theta, \qquad d = \frac{h}{\tan\theta}$
The sun's elevation and a shadow
Sun's elevation from a shadow
$\tan\theta = \frac{\text{height}}{\text{shadow}}$

Watch out for (6)

Measuring depression from the tower
→ Line of sight, elevation and depression
Putting the depression angle at the top of the triangle
→ Line of sight, elevation and depression
Turning tan upside down
→ Height and distance through tan
Dividing when you should multiply
→ Height and distance through tan
Reading the ratio the wrong way
→ The sun's elevation and a shadow
A high sun does not lengthen the shadow
→ The sun's elevation and a shadow

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