CLASS 10 MATHEMATICS • HEIGHTS & DISTANCES
Some Applications of Trigonometry
Worksheet with Online Quiz
Turn a word problem into a clear diagram. Explore towers, rooftops, ladders and boats with interactive practice that keeps the question and its model together.
Read the Diagram
Locate the horizontal, vertical height and line of sight.
Explore the models →Choose Your Ratio
Connect real measurements with the correct trigonometric relationship.
Solve heights and distances →Think in Two Triangles
Use a shared height to compare two observation points.
Build exam confidence →Applications of Trigonometry Class 10 Online Quiz
Question 1 of 20 • Easy • Answered 0/20
Looking above eye-level horizontal gives an angle of…
Quiz instructions
20 questions: 10 easy, five average and five HOTS. The timer starts when you answer or navigate. Choosing an option opens the next question. Use the numbered buttons to revisit answers. Each correct answer earns one mark; there is no negative marking. Review solutions after submitting or when time ends.
Heights and Distances Online Games
Spot the Measurement
Tap a coloured line or choose its label below.
Select the vertical height in this ground-level tower model.
How to play
Choose from five games using the raised selector. Tap diagram lines, adjust a survey angle, match formulas, sort statements or arrange a solution. Every game contains five tasks.
Applications of Trigonometry Online Activities
Answer entry help
Use exact fractions or radicals: 7/2, 8√3, 8*sqrt(3), or 6*(sqrt(3)+1). If using decimals, give at least three decimal places. Enter measurements without the unit; the field label provides it. Answers remain available while this page stays open.
Extra Practice: Shadows, Buildings and Case-Based Questions
Sun and Shadow Practice
A vertical pole and its shadow form a right triangle on level ground.
The shadow is 14 m long and the Sun’s elevation is 30°. Find pole height.
Enter exact fractions or radicals, such as 8√3 or 8*sqrt(3). For decimals, use at least three decimal places.
About these practice sets
Five sets contain five tasks each. Practise shadows, building comparisons, speed from two observations, survey cases and creating valid examples. Write your own diagram and equations before checking.
Student Tips: Solve Heights and Distances with Confidence
- Underline the measurement you need to find.
- Draw the horizontal through the observer’s eye before marking an angle.
- Show the right angle between the vertical object and level ground.
- Distinguish horizontal distance from a ladder, string or line-of-sight length.
- Use H−e when the observer’s eyes are e metres above ground.
- For two positions, use the same height in both equations.
- On the same side subtract horizontal distances; for opposite sides add them.
- Keep radicals exact until the final step and include units in written answers.
Teaching Tips for Parents and Teachers
- Ask learners to point to the horizontal before naming elevation or depression.
- Use a toy tower and string to distinguish height from slope length.
- Let students predict how the angle changes before moving the slider.
- Ask what stays constant when the observer changes position.
- Discuss an incorrect diagram as well as an incorrect calculation.
Some Applications of Trigonometry Class 10: Notes, Formulas and Solved Questions
This Some Applications of Trigonometry Class 10 worksheet develops the skill of translating a situation into one or two right triangles. The interactive quiz checks concepts and calculations, while the games focus on diagrams, ratio selection and explaining a method. Use the worksheet download option already provided on this page for written work.
Angle of Elevation and Angle of Depression: Class 10 Explanation
The line of sight joins the observer’s eye to the point being viewed. Measure an angle of elevation upwards from the horizontal through the eye. Measure an angle of depression downwards from that horizontal. For the same sight line between two points, the corresponding elevation and depression angles are equal when their horizontal reference lines are parallel.
Some Applications of Trigonometry Class 10 Formula Sheet
Let h be the vertical difference, d the horizontal distance and L the straight sloping length. Angle θ is measured from horizontal.
| Information | Relationship | Use |
|---|---|---|
| Height and distance | tan θ = h/d | h = d tan θ |
| Distance from height | d = h/tan θ | Ground gap or shadow |
| Vertical and sloping lengths | sin θ = h/L | h = L sin θ |
| Horizontal and sloping lengths | cos θ = d/L | d = L cos θ |
| Observer’s eye height e | H = e + d tan θ | Top above eye level on level ground |
Standard Values for Heights and Distances
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
Applications of Trigonometry Class 10 Important Questions
Revise these useful question types: a tower from one point, a ladder or taut string, observer eye height, an angle of depression, a shadow, two positions on the same side, points on opposite sides, and a building with a mast. For each type, identify the right triangles and explain why the chosen ratio uses the known measurements. “Important” describes revision coverage rather than a prediction of a future paper.
Some Applications of Trigonometry Class 10 Examples with Solutions
Example 1: Tower height from one ground point
A tower’s top is viewed at 60° from level ground 17 m from its foot. Find its height.
tan 60° = h/17
√3 = h/17
h = 17√3 m
Example 2: Include the observer’s eye height
Eyes are 1.6 m above level ground. From 21 m away, the top of a building is seen at 45°. Find its full height H.
tan 45° = (H−1.6)/21
1 = (H−1.6)/21
H−1.6 = 21
H = 22.6 m
Example 3: Two observation points on the same side
A ground observer moves 26 m towards a tower. Elevation changes from 30° to 60°. Find tower height.
Let x be the nearer distance and h the shared height.
h = x tan 60°
h = x√3
h = (x+26) tan 30°
h = (x+26)/√3
x√3 = (x+26)/√3
3x = x+26
2x = 26
x = 13
h = 13√3 m
Applications of Trigonometry Class 10 Case-Based Questions
Survey a school flagpole
A surveyor stands 14 m from a vertical flagpole on level ground. The eye height is 1.5 m and elevation is 45°.
1. Which ratio is useful? Tangent, because it connects the vertical difference with the horizontal distance.
2. What is the height above eye level?
h = 14 tan 45°
h = 14 m
3. What is the full flagpole height?
H = 1.5 + 14
H = 15.5 m
Some Applications of Trigonometry Class 10 Extra Questions with Answers
- A 34 m cliff has a ground marker at depression 45°. Find horizontal distance.
Show solution
d = 34/tan 45°
d = 34 m
- A 32 m ladder makes 30° with ground. Find its vertical reach.
Show solution
h = 32 sin 30°
h = 16 m
- A 12 m pole casts a shadow at Sun elevation 60°. Find shadow length.
Show solution
d = 12/tan 60°
d = 4√3 m
- From a 10 m roof, a taller roof is seen at elevation 45°, across a 17 m level horizontal gap. Find taller building height.
Show solution
H = 10 + 17 tan 45°
H = 27 m
- A tower 15 m high stands between two ground observers whose elevations are 30° and 60°. Find their separation.
Show solution
d₁ = 15/tan 30°
d₁ = 15√3
d₂ = 15/tan 60°
d₂ = 5√3
Separation = 20√3 m
NCERT Exercise 9.1: How to Approach Word Problems
Read the wording and sketch first. Label the unknown, mark the horizontal and right angle, choose a ratio, substitute exact standard values and solve one step per line. Check the question text against your textbook edition when using an Exercise 9.1 solution. The practice here is original and is not a complete reproduction of the NCERT exercise.
Current Syllabus and Official Sample-Paper Update
The official CBSE Mathematics syllabus for 2026–27 specifies simple heights-and-distances problems using no more than two right triangles, with elevation or depression angles of 30°, 45° and 60°. The 12 marks shown for the Trigonometry unit cover both trigonometry chapters, rather than this chapter alone.
The current official Mathematics Standard sample paper includes a kite-height problem using a given tangent ratio and string length. This supports practising how a ratio identifies proportional sides before using the hypotenuse. It is an official sample-paper example, not evidence of an exact question repeatedly appearing in board exams. Use the official marking scheme alongside it.
Common Heights and Distances Mistakes
Avoid measuring depression from vertical, treating the string as horizontal distance, forgetting eye height or adding distances for two positions on the same side. A clear labelled diagram often exposes these errors before the calculation begins.
Related Class 10 Maths Worksheets and Notes
Some Applications of Trigonometry Class 10 FAQs
How do I solve heights and distances questions in Class 10?
Draw a labelled right triangle, mark the angle from horizontal and identify the known and unknown sides. Choose tan, sin or cos to connect them, substitute exact values and solve. Use two equations when two triangles share a height.
What is the difference between elevation and depression?
Elevation is measured upwards from the observer’s horizontal. Depression is measured downwards from it. Neither is measured directly from vertical.
Why is angle of depression equal to angle of elevation?
For the same sight line between two points, their horizontal lines are parallel. The sight line is a transversal, so the relevant alternate interior angles are equal.
When should I use tan, sin or cos in word problems?
Use tan for vertical difference and horizontal distance, sin for vertical difference and sloping length, and cos for horizontal distance and sloping length. Choose using the sides involved rather than the name of the object.
When do I add the observer’s height?
If the eye is e above level ground and the top is above eye level, tangent gives H−e. Add e to obtain full height H. If the observer is on a roof, use that observation height instead.
Is a kite string the height or the hypotenuse?
A straight taut string is the hypotenuse of the working triangle. Multiply its length by sin θ for vertical rise. Add the hand’s ground height when needed to obtain the kite’s full height.
How do I solve two observation point problems?
Choose one shared height h. If the nearer distance is x and movement towards the object is s, the farther distance is x+s. Write a tangent equation for each position and equate the two expressions for h.
When should I add or subtract two horizontal distances?
For collinear observation points on the same side of the object, subtract near distance from far distance. For observers on opposite sides with the object between them, add their distances from its foot.
Which Applications of Trigonometry questions are important for exams?
Cover one-point towers, ladders and strings, eye height, depression angles, shadows and two-triangle problems. Practise diagrams and explanations as well as final numbers. Refer to verified official papers for actual exam examples.
How do I calculate a pole’s height from its shadow?
On level ground, h = shadow length × tan(Sun elevation). The shadow is horizontal and the pole is vertical. A higher Sun angle gives a shorter shadow for the same pole.
Does this chapter have a fixed 12-mark weightage?
The current CBSE course structure assigns 12 marks to the whole Trigonometry unit, including Introduction to Trigonometry and Heights and Distances. It does not guarantee 12 marks for this chapter alone.
How should I present an answer in a board-style question?
Include a labelled figure, defined variables, the ratio equation, exact substitution, clear working and a final statement with units. Use the question’s marking scheme to check which steps earn credit.
Can You Turn the Story into a Triangle?
Read the situation, choose the ratio and test your progress.
Try the Online Quiz
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