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Heights and distances is where Class 10 trigonometry meets the real world: towers, kites, lighthouses and boats. The maths is usually one or two lines of tan. What goes wrong is the figure: the angle at the wrong corner, the observer's height forgotten, or two triangles drawn as one. This post sets out the syllabus limits, a way to draw every figure, three worked problems and the slips to avoid.
What does the syllabus allow?
CBSE's Class X Mathematics syllabus for 2026-27 lists angle of elevation and angle of depression, with "simple problems on heights and distances". It sets two limits: problems "should not involve more than two right triangles", and angles of elevation or depression "should be only 30°, 45°, and 60°". So you need the values of tan at those three angles, and the sides of the 30-60-90 and 45-45-90 triangles. The NCERT chapter, Some Applications of Trigonometry, reprinted for 2026-27, has seven worked examples and one exercise.
What are the angles of elevation and depression?
Angle of elevation: you look up at an object. It is the angle between the horizontal line from your eye and your line of sight.
Angle of depression: you look down at an object. It is the angle between the horizontal line from your eye and your line of sight.
Both are measured from the horizontal at the observer, never from the vertical. Because the horizontal at the top and the ground are parallel, the angle of depression from the top of a tower to a boat equals the angle of elevation from the boat to the top of the tower (alternate angles). That fact puts the angle inside your right triangle.
Why does a simple sketch help?
Hegarty and Kozhevnikov (1999) separated two kinds of mental images used in maths problems: schematic ones, which encode the spatial relations in a problem, and pictorial ones, which encode what the objects look like. Using schematic representations was associated with success in problem solving, while pictorial ones were negatively correlated with it. A stick-figure triangle with labels is the schematic kind.
Worked problems
Useful values: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3. Take √3 ≈ 1.732 only at the end, if the question asks.
Problem 1. One triangle. From a point on the ground 30 m from the foot of a tower, the angle of elevation of the top is 60°. Find the height.
Figure: tower AB of height h, point C on the ground with BC = 30 m, angle ACB = 60°.
tan 60° = AB/BC, so √3 = h/30.
h = 30√3 m ≈ 51.96 m.
Problem 2. Angle of depression. From the top of a lighthouse 75 m high, the angle of depression of a boat is 45°. How far is the boat from the foot of the lighthouse?
The angle of elevation of the top from the boat is also 45°.
tan 45° = 75/d, so 1 = 75/d.
d = 75 m.
Problem 3. Two triangles. From a point on the ground, the angle of elevation of the top of a tower is 30°. After walking 40 m towards the tower, it is 60°. Find the height of the tower.
Let the height be h and the distance from the second point to the foot be x. The first point is x + 40 from the foot.
Second point: tan 60° = h/x, so h = x√3.
First point: tan 30° = h/(x + 40), so h = (x + 40)/√3.
Set them equal: x√3 = (x + 40)/√3. Multiply both sides by √3: 3x = x + 40.
2x = 40, so x = 20 m, and h = 20√3 m ≈ 34.64 m.
Check: tan 30° should be h/(x + 40) = 20√3/60 = √3/3 = 1/√3. It is.
Which mistakes lose marks?
Angle at the wrong corner. Angle of depression is at the observer, from the horizontal. Move it into the triangle using alternate angles.
Forgetting the observer's height. If eye level is 1.5 m above the ground, the triangle gives the height above eye level. Add 1.5 m at the end.
Using sin when you need tan. If the question gives a horizontal distance and asks a height, or the reverse, it is tan. Use sin or cos only when a slant length, such as a ladder or kite string, is involved.
Merging two triangles. In Problem 3, write one equation per triangle, then combine.
Rounding too early. Keep √3 until the last line.
No figure. Marking schemes often give a mark for the figure alone.
If you can follow solutions but cannot set up the second triangle yourself, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning. And when it helps to see an idea move rather than read it, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read.
A one-week practice routine
Day 1: Six figures only, no solving. Read each NCERT exercise question and draw the schematic.
Day 2: Five one-triangle problems with elevation and depression mixed.
Day 3: Three problems with the observer's height. Write the "add eye level" step every time.
Day 4: Three two-triangle problems like Problem 3. Check each answer in both triangles.
Day 5: A timed set of five, then redraw any figure you got wrong.
If the values or ratios feel shaky, start with introduction to trigonometry in Class 10. For how the board paper marks figures, see common Class 10 maths board mistakes.
Today, take three questions from the NCERT exercise and draw only the figures, with every angle in the right corner.
