Question
A man standing on the ground observes the top of a building at an angle of elevation of 60°. If the man is standing 40 metres away from the foot of the building, find the height of the building.
Solution — Step by Step
Height and distance problems always start with a diagram. Drawing it correctly is half the work.
Step 1: Draw and label the diagram.
- Let B = the base (foot) of the building.
- Let T = the top of the building.
- Let M = the man's position.
The man M is standing 40 m from the base B, so MB = 40 m. The angle of elevation from M to T is 60°, so ∠TMB = 60°. The angle at B is 90° (building is vertical, ground is horizontal). TB = height of the building = h (what we need).
Step 2: Identify the right triangle.
Triangle TBM is right-angled at B (the building meets the ground at 90°).
In this right triangle:
- Angle at M = 60°
- Side opposite to 60° = TB = h (the height, what we want)
- Side adjacent to 60° = MB = 40 m (the distance, what we know)
Step 3: Set up the trigonometric equation.
We have the opposite and adjacent sides relative to the 60° angle. That's the tangent ratio:
Key Relationship
tan(angle of elevation) = Height / Base distance
tan 60° = h / 40
Step 4: Substitute the standard value and solve.
tan 60° = √3
So: √3 = h / 40
h = 40√3 metres
Step 5: Approximate if needed.
40√3 ≈ 40 × 1.732 = 69.28 metres
For board exams, leave the answer as 40√3 m unless the question asks for a decimal approximation.
Answer
Height of the building = 40√3 metres ≈ 69.28 m
Why This Works
An angle of elevation is measured upward from the horizontal line of sight. When you stand on flat ground and look up at something, the horizontal, the vertical (height), and your line of sight form a right triangle. The tangent ratio connects the vertical (opposite) and horizontal (adjacent) sides, making it the natural choice for height-distance problems.
Alternative Method: Using Sine
If the question gave us the distance along the line of sight (hypotenuse) instead of the base, we'd use sine:
sin θ = h / (line of sight distance)
Here we have the base, so tan is simpler. Always check which two quantities are given before picking a trig ratio.
💡 Expert Tip
For every height-distance problem: identify what you know (which side) and what you want (which side). Then check:
- Know adjacent, want opposite → use tan
- Know hypotenuse, want opposite → use sin
- Know hypotenuse, want adjacent → use cos
This simple check prevents picking the wrong ratio.
Common Mistake
⚠️ Common Mistake
Mistake: Setting tan 60° = 40/h instead of h/40.
tan = opposite/adjacent. The height (what we're finding) is the opposite side to the angle of elevation. The base distance (40 m) is the adjacent side. Writing it upside down gives h = 40/√3 instead of 40√3 — a completely different answer. Always label the triangle sides relative to the given angle before writing the trig ratio.