Heights and distances: Class 10 practice
The angle of elevation is measured up from the horizontal to an object above. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Trigonometric ratios of 30°, 45°, 60°
The idea behind heights and distances
The angle of elevation is measured up from the horizontal to an object above. With the horizontal distance and the angle, height = distance × tan(angle).
Step-by-step method
- Draw the right triangle: ground, height, line of sight.
- tan θ = height ÷ distance.
- Use tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
- Give exact and approximate (√3 ≈ 1.732) answers.
Worked example 1 (Basic)
From a point 95 m from the foot of a tower, the angle of elevation of its top is 45°. Find the height of the tower.
- tan 45° = h ÷ 95
- h = 95 × tan 45° = 95 m
- h ≈ 95.00 m
- Answer: 95 m ≈ 95.00 m
Worked example 2 (Standard)
From a point 70 m from the foot of a tower, the angle of elevation of its top is 60°. Find the height of the tower.
- tan 60° = h ÷ 70
- h = 70 × tan 60° = 70√3 m
- h ≈ 121.24 m
- Answer: 70√3 m ≈ 121.24 m
Worked example 3 (Challenge)
From a point 70 m from the foot of a tower, the angle of elevation of its top is 30°. Find the height of the tower.
- tan 30° = h ÷ 70
- h = 70 × tan 30° = 70/√3 = (70/3)√3 m
- h ≈ 40.41 m
- Answer: 70/√3 = (70/3)√3 m ≈ 40.41 m
Common mistakes and misconceptions
- Using sin instead of tan
- Measuring the angle from the vertical
Practice questions
- (Basic) From a point 80 m from the foot of a tower, the angle of elevation of its top is 45°. Find the height of the tower.
- (Basic) From a point 90 m from the foot of a tower, the angle of elevation of its top is 45°. Find the height of the tower.
- (Standard) From a point 30 m from the foot of a tower, the angle of elevation of its top is 60°. Find the height of the tower.
- (Standard) From a point 80 m from the foot of a tower, the angle of elevation of its top is 60°. Find the height of the tower.
- (Challenge) From a point 40 m from the foot of a tower, the angle of elevation of its top is 30°. Find the height of the tower.
- (Challenge) From a point 25 m from the foot of a tower, the angle of elevation of its top is 30°. Find the height of the tower.
Answer key with full working
- Q1: tan 45° = h ÷ 80 → h = 80 × tan 45° = 80 m → h ≈ 80.00 m → Answer: 80 m ≈ 80.00 m
- Q2: tan 45° = h ÷ 90 → h = 90 × tan 45° = 90 m → h ≈ 90.00 m → Answer: 90 m ≈ 90.00 m
- Q3: tan 60° = h ÷ 30 → h = 30 × tan 60° = 30√3 m → h ≈ 51.96 m → Answer: 30√3 m ≈ 51.96 m
- Q4: tan 60° = h ÷ 80 → h = 80 × tan 60° = 80√3 m → h ≈ 138.56 m → Answer: 80√3 m ≈ 138.56 m
- Q5: tan 30° = h ÷ 40 → h = 40 × tan 30° = 40/√3 = (40/3)√3 m → h ≈ 23.09 m → Answer: 40/√3 = (40/3)√3 m ≈ 23.09 m
- Q6: tan 30° = h ÷ 25 → h = 25 × tan 30° = 25/√3 = (25/3)√3 m → h ≈ 14.43 m → Answer: 25/√3 = (25/3)√3 m ≈ 14.43 m
Syllabus note
This skill is listed under "Some Applications of Trigonometry" for Class 10 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is heights and distances in?
"Some Applications of Trigonometry", Class 10 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.