Heights and Distances – Complete Guide With Examples
Heights and Distances is one of the most practical applications of trigonometry in mathematics. It helps us calculate the height of tall objects like buildings, towers, mountains, and trees without actually measuring them directly.
In Class 10 mathematics, this topic is introduced as part of trigonometry. By using trigonometric ratios such as sine, cosine, and tangent, we can determine unknown heights and distances when certain angles and lengths are known.
This complete guide explains the concept of heights and distances with diagrams, important formulas, step-by-step solved examples, and exam tips to help students prepare for CBSE board exams and competitive exams.
What Are Heights and Distances?
Heights and distances problems involve finding the height of an object or the distance between objects using angles of elevation or angles of depression.
These problems are solved using right-angled triangles and trigonometric ratios.
- Height: The vertical measurement of an object.
- Distance: The horizontal distance between two points.
- Angle of Elevation: The angle formed when looking up at an object.
- Angle of Depression: The angle formed when looking downward at an object.
Angle of Elevation
When an observer looks at an object above eye level, the angle formed between the horizontal line and the line of sight is called the angle of elevation.
Example: Looking up at the top of a building from the ground.
Angle of Depression
When an observer looks downwards at an object below eye level, the angle formed between the horizontal line and the line of sight is called the angle of depression.
Example: Looking down from a lighthouse towards a boat in the sea.
Trigonometric Ratios Used in Heights and Distances
Three main trigonometric ratios are used in solving heights and distances problems:
- sin θ = Opposite / Hypotenuse
- cos θ = Adjacent / Hypotenuse
- tan θ = Opposite / Adjacent
In most heights and distances questions, the tangent ratio is commonly used.
tan θ = Height / Distance
Basic Formula for Heights and Distances
The fundamental formula used in most problems is:
tan θ = Opposite Side / Adjacent Side
In real-life problems:
- Opposite side = Height of object
- Adjacent side = Distance from observer
Solved Heights and Distances Problems
Problem 1: Finding Height of a Tower
Question: The angle of elevation of the top of a tower from a point on the ground is 45°. If the distance from the point to the tower is 20 m, find the height of the tower.
Solution:
Distance = 20 m Angle = 45°
tan 45° = Height / 20
1 = Height / 20
Height = 20 m
Height of the tower = 20 meters
Problem 2: Finding Distance From a Building
Question: The angle of elevation of the top of a building is 30°. If the height of the building is 15 m, find the distance of the observer from the building.
Solution:
tan 30° = Height / Distance
1/√3 = 15 / Distance
Distance = 15√3 ≈ 26 m
Distance from building ≈ 26 meters
Problem 3: Height of a Tree
Question: The angle of elevation of the top of a tree from a point 10 m away is 60°. Find the height of the tree.
Solution:
tan 60° = Height / 10
√3 = Height / 10
Height = 10√3 ≈ 17.32 m
Height of the tree ≈ 17.32 meters
Problem 4: Lighthouse and Boat
Question: The angle of depression from the top of a lighthouse to a boat in the sea is 30°. If the lighthouse is 40 m high, find the distance of the boat from the lighthouse.
Solution:
Angle of depression = Angle of elevation = 30°
tan 30° = 40 / Distance
1/√3 = 40 / Distance
Distance = 40√3 ≈ 69.28 m
Distance ≈ 69 meters
Problem 5: Two Angles of Elevation
Question: The angle of elevation of the top of a tower from two points on the same straight line and on the same side of the tower are 30° and 60° respectively. If the distance between the points is 10 m, find the height of the tower.
Solution (Concept):
Let height = h
Using tan formulas:
tan 60° = h / x
tan 30° = h / (x + 10)
Solving these equations gives:
Height ≈ 8.66 m
Real Life Applications of Heights and Distances
Heights and distances concepts are used in many real-world fields:
- Architecture and construction
- Surveying land and measuring mountains
- Aviation and navigation
- Engineering projects
- Satellite measurements
Exam Tips for Solving Heights and Distances Problems
✔ Identify opposite and adjacent sides correctly
✔ Use correct trigonometric ratio
✔ Use trigonometric value table for standard angles
✔ Always write units in the final answer
Common Mistakes Students Make
- Confusing angle of elevation and depression
- Using incorrect trigonometric ratio
- Forgetting to convert angles correctly
- Calculation errors in square roots
Quick Revision Table
- tan θ = Height / Distance
- sin θ = Opposite / Hypotenuse
- cos θ = Adjacent / Hypotenuse
- Angle of elevation → looking upward
- Angle of depression → looking downward
Conclusion
Heights and distances is a powerful application of trigonometry that allows us to measure objects that are difficult to measure directly. By understanding angles of elevation, angles of depression, and basic trigonometric ratios, students can solve these problems easily.
Regular practice of solved examples is the best way to master this topic and score full marks in the Class 10 mathematics examination.