Angles of elevation and depression
Heights and distances
- In 5 of 20 recent exams
- Paper 2, calculator
- 1 to 6 marks
How it is set in the exam
Paper 2 (calculator) only, 1–6 marks a part, usually the opening parts of a longer context question: a windmill, a cone seen from beyond its base, two buildings seen from a point between them. Typical (5 of 20 exam versions, 10 parts).
Exam phrases to spot (4)
“the angle of elevation of the top of the tower is …”
“the angle of depression of the boat from the cliff top”
“find the height of the building”
“from a point further along the road”
In disguise (3)
- two buildings or masts with angles marked from the horizontal at the top of one of them, and no elevation or depression wording
- an observer standing beyond the edge of a cone or pyramid, looking at its top
- a line of sight given by its angle with a vertical wall: its angle from the horizontal is 90°−α
Variants (5)
- one right triangle: a height, a distance or an angle, in degrees or in radians
- a vertical made of pieces: eye level, a flagpole on a roof, a known extra length
- two sightings in line with the foot: two tan equations, eliminate the height
- two verticals: a horizontal line from one top splits the other vertical into two tan lengths
- depression from a height given in metres, with the distance wanted in kilometres
Looks like it, but isn't (3)
- the angle between an edge of a solid and its base (no observer, nothing measured from outside): Triangles inside a solid
- a right triangle already drawn with its interior angles, and nobody looking at anything: Find a side or an angle: choose the rule
- a boat's position given by bearings: Bearings (a height seen from the boat, in the same question, stays here)
A vertical in pieces: eye level and a flagpole
A flagpole stands on the flat roof of a building, at its front edge. Chen stands on level ground 48 m from the foot of the building. Using an instrument held 1.4 m above the ground, she measures the angles of elevation of the bottom and the top of the flagpole as 31° and 36°.
(a) Find the height of the building. [3]
(b) Find the length of the flagpole. [3]
Try it yourself first, then compare.
Now try one like it
Write your working on paper or on screen. Paper Zero marks it step by step, like the worked example, and shows where each mark went. Free, with no account.
- 1
Draw the horizontal
at the observer's eye and the vertical at the object. Mark each angle from the horizontal: put a depression angle at the lower end (alternate angles), and turn an angle to the vertical into 90°−α. For an observer on the ground between two verticals, in line with their feet, the two angles of elevation and the angle between the two lines of sight fill the straight line of the ground at the observer, so they add up to 180°.
- 2
Find the right triangles:
each has part of the vertical as one side and a horizontal distance as the other.
- 3
One triangle:
tan for a height or a distance (sin or cos when the line of sight's length is involved).
- 4
Two triangles sharing the height:
one tan equation each, then eliminate the height (solve simultaneous linear equations); or use the sine rule in the triangle between the two lines of sight.
- 5
Add or subtract
the known lengths (eye level, the other pieces of a vertical), and check the units.
The angle measured from the vertical,
or from the line of sight, instead of from the horizontal.
A depression angle used as the angle inside the triangle at the top
at the top, the angle inside the right triangle is 90°−α. Move the depression angle to the lower end (alternate angles), where it is α.
Eye level forgotten
when a height above the ground is asked, or added when a difference of heights is asked (it cancels).
More mistakes (3)
Radians
an angle with no degree sign is in radians. 25tan0.7=21.1, but in degree mode the same keys give 0.305: a result far too small for the context is the sign (check the settings).
Units
a height in metres and a distance wanted in kilometres: convert first.
Rounding x before finding the height
in a two-sighting question.