Angle of Elevation and Depression

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Angle of Elevation and Depression: meaning, method and solved example (Geometry) – Maths PDF

In short: An angle of elevation is measured upward from the horizontal to a line of sight, while an angle of depression is measured downward from the horizontal to an object below.

GeometryTrigonometryG-SRT.C.8

What is Angle of Elevation and Depression?

Look straight ahead, then tilt your eyes up toward the top of a tower: the opening between that level gaze and the new line is the angle of elevation. Tilt your eyes down toward a boat instead and you create an angle of depression.

Both are measured from the horizontal, never from the vertical. One handy fact: the depression from a tower top down to a boat equals the elevation from that boat up to the tower top, because the two horizontal lines are parallel and the angles are alternate interior angles.

How to spot it

  • A story mentions looking up at a flagpole, plane, kite, cliff edge or balloon with a stated tilt.
  • A story mentions looking down from a tower, roof or airplane at a boat, car or camp below.
  • A right triangle hides in the scene, with the ground distance as one leg and the height difference as the other.
  • The words line of sight, eye level or horizontal appear in the statement.

The method

  1. Sketch the scene, draw a horizontal line from the eye, and place the tilt between it and the line of sight.
  2. Mark the right triangle: ground distance, height difference and line of sight.
  3. Use tangent when both legs are involved, and sine or cosine when the line of sight is involved.
  4. Solve for the unknown, and add the observer’s eye height if the problem says it matters.
25° elevation25° depressionabout 140 ft300 ftboat
Elevation from the boat and depression from the lamp are both 25°, measured from horizontal lines.

Angle of Elevation and Depression: solved example

Problem

From a fishing boat 300 feet from the foot of a lighthouse, the angle of elevation to the lamp is \(25^{\circ}\). How tall is the lighthouse above the water, and what is the angle of depression from the lamp to the boat? Use \(\tan 25^{\circ} \approx 0.4663\).

Solution

The 300 feet is the leg beside the tilt, and the height is the leg across from it, so \(\tan 25^{\circ} = \frac{h}{300}\). Then \(h = 300 \times 0.4663 \approx 139.9\).

The horizontal lines at the boat and at the lamp are parallel, so the depression equals the elevation. The lighthouse is about 140 feet tall, and the angle of depression is \(25^{\circ}\).

Try it

Question 1

You stand 80 feet from a building and see its roofline at \(51^{\circ}\) above level. Ignoring your eye height, how tall is it? Use \(\tan 51^{\circ} \approx 1.2349\).

Show the answer

\(80 \times 1.2349 \approx 98.8\) feet.

Question 2

From a cliff 120 meters above the sea, a ship appears at an angle of depression of \(20^{\circ}\). How far is the ship from the cliff base? Use \(\tan 20^{\circ} \approx 0.3640\).

Show the answer

The depression equals the elevation from the ship, so \(\tan 20^{\circ} = \frac{120}{d}\) and \(d = \frac{120}{0.3640} \approx 329.7\) meters.

Angle of Elevation and Depression vs. Tangent

The angle is the thing you measure: a tilt from the horizontal up or down to a line of sight. The tangent ratio is the tool you use to calculate with it, opposite over adjacent in the right triangle formed. A problem names the tilt in degrees; the ratio turns that tilt into a height or a distance.

Tangent (Trigonometric Ratio)

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