Introduction
Height & Distance applies the trigonometric ratios from Chapter 25 to real-world scenarios — towers,
buildings, ships — using two key concepts: angle of elevation and angle of depression.
Angle of Elevation vs Angle of Depression
Flowchart — Elevation vs Depression
Angle of Elevation — angle formed looking UP from the horizontal to an object
Angle of Depression — angle formed looking DOWN from the horizontal to an object
By the alternate-angle property of parallel lines, the angle of elevation from the ground to a point equals the angle of depression from that point back to the ground.
Standard Setup
In almost every question: Height = opposite side, Distance (horizontal) = adjacent side → so tan θ = Height / Distance is the formula used in the large majority of problems.
Q. The angle of elevation of the top of a tower from a point on the ground 30 m away is 30°. Find the height of the tower.
tan30° = Height/30
1/√3 = Height/30
Height = 30/√3 = 10√3 m
"Two Angles, Find Distance/Height" Pattern
Q. From the top of a tower 100 m high, the angles of depression of two points on the same side on the ground are 30° and 60°. Find the distance between the two points.
For 60° angle: tan60° = 100/d₁ → d₁ = 100/√3
For 30° angle: tan30° = 100/d₂ → d₂ = 100√3
Distance between points = d₂ − d₁ = 100√3 − 100/√3 = 100(3−1)/√3 = 200/√3 = 200√3/3 m
"Height of Tower from Two Points" Pattern
Q. A man standing 40 m from a tower observes the angle of elevation to be 30°. He walks towards the tower and at a point observes the angle to be 60°. How far did he walk?
Let height = h. At 40 m: tan30° = h/40 → h = 40/√3
At new distance d: tan60° = h/d → d = h/√3 = 40/3
Distance walked = 40 − 40/3 = 80/3 m
Height of Two Objects — Ratio Pattern
💡 Exam Tip: When a question gives angles of elevation from the same point to two objects of different heights, set up tan θ = height/distance for each object separately, then combine (add, subtract, or take a ratio) as the question requires — don't try to combine the two triangles into one equation prematurely.
✅ Practice Focus: Elevation vs depression definitions and their equality via alternate angles · tan θ = height/distance as the default formula · Two-angle "distance between points" pattern · Standard angle values (30°,45°,60°) applied directly from Chapter 25.