Angle of elevation

The angle of elevation is the angle formed between the horizontal line and the line of sight directed upward toward an object. It is measured from the horizontal plane and indicates the height you need to look upward to see objects such as a building top or a flying kite. It is a key concept in trigonometry and is widely used in surveying, navigation, and architecture.

Table of Content 

• Angle of Elevation Definition
• Angle of Elevation Formula
• Angle of Elevation and Angle of Depression
• Angle of Elevation Examples

Angle of Elevation Definition

The angle formed between the horizontal plane and the line of sight when an observer looks upward toward an object above the horizontal level.

Key attributes of this angle

• It is measured from the horizontal line upward

• It is always between 0° and 90°

• It originates at the observer's eye level

• It is part of a right triangle in trigonometric problems

Angle of Elevation Formula

The angle of elevation connects three values  the height of the object (opposite side), the horizontal distance from the observer to the base (adjacent side), and the line of sight (hypotenuse). These are related through the standard trigonometric ratios.

Let:

• θ = angle of elevation

• h = vertical height of the object

• d = horizontal distance from observer to base

The three primary formulas are:

The most commonly used formula is:

 tan⁡(θ)=Opposite (height)Adjacent (distance)

Which rearranges to:

• h = d × tan θ  to find height

• d = h / tan θ  to find horizontal distance

• θ = tan⁻¹(h / d)  to find the angle

Angle of Elevation And Angle of Depression

The Angle of Elevation And Angle of Depression. These two angles are closely related but operate in opposite directions.

Alternate Angle Property: When a person on the ground looks up at an object, and a person at the top looks down, both angles are equal. This happens because the horizontal lines are parallel, and the line of sight forms equal alternate interior angles.

Angle of Elevation Examples

Example 1: A person standing 40 m away from a tree observes its top at an angle of elevation of 35°. Find the height of the tree.

Solution:

Using: tan θ = h / d

tan 35° = h / 40

h = 40 × tan 35°

h = 40 × 0.7002

h = 28.01 m

Example 2: An observer is standing 80 m from the base of a building. The building is 80 m tall. What is the angle of elevation to the top?

Solution:

Using: θ = tan⁻¹(h / d)

θ = tan⁻¹(80 / 80)

θ = tan⁻¹(1)

θ = 45°

Example 3: Finding Height of a Tower

Solution:

A person stands 10 m away from a tower. The angle of elevation to the top is 30°. Find the height of the tower.

tanθ=heightdistance

tan⁡30∘=h10

13=h10

h=103≈5.77m

Answer: Height ≈ 5.77 m.

Related Topics

• Height and Distance Problems In Trigonometry
• Application Problems Trigonometry
• Applications of Trigonometry
• Trigonometry Exercises with Answers
• Applications of Trigonometry: Important Questions and Answers for Class 10
• Trigonometry: Important Questions and Answers for Class 10