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Please enter the height of the object & your horizontal distance from it to calculate the angle of elevation.

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Use this angle of elevation calculator to determine the angle between the horizontal line of sight and the object up above it. It uses measurements (height and horizontal distance) in order to perform the calculation for the angle of elevation.

With this calculator, users can easily calculate:

- Angle of Elevation (θ)
- Horizontal distance (rise)
- Vertical height (run)
- Elevation grade & its percentage

**“The angle of elevation is the measurement of the angle formed between a horizontal line of sight and the line of sight to an object located above the horizontal line”**

It's the amount of upward tilt needed from your eyes to see the object. This measurement is valuable in various fields like surveying, astronomy, navigation, etc., helping to determine an object's height or the distance between the observer and the object.

You can calculate the elevation angle using an angle of elevation formula:

\(\text{Angle of Elevation} = tan^{-1}\left(\dfrac{height}{\text{horizontal distance}}\right)\) or \(AOE = tan^{-1}\left(\dfrac{h}{d}\right)\)

What would be the value of the angle of elevation for the above figure?

**Solution:**

**Data Given:**

- Horizontal Distance = 290 ft
- Vertical height = 134 ft

**Calculations:**

Using the angle of elevation formula:

\(\text{Angle of Elevation} = tan^{-1}\left(\dfrac{height}{\text{horizontal distance}}\right)\)

\(\text{Angle of Elevation} = tan^{-1}\left(\dfrac{134 ft}{290 ft}\right)\)

\(\text{Angle of Elevation} = tan^{-1}\left(0.462\right)\) \(\text{Angle of Elevation} = 24.8^\text{o}\)

Converting to radians:

\(\ Radians = Degrees × \dfrac{π}{180^\text{o}}\)

\(\ Radians = 24.8^\text{o} × \dfrac{3.14}{180^\text{o}}\)

\(\ Radians = 24.8^\text{o} × 0.017\) \(Radians = 0.4216\)

Suppose you are standing at point ‘P’ and observing a bird sitting point ‘Q’ at an elevation angle of \(45^{o}\). Now if the building is about 10 meters away from your position, what would be its maximum height?

**Solution:**

**Data Given:**

- AOE = ∠QPR = \(45^{o}\)
- Horizontal Distance = d = 10 meters

**Calculations:**

Using vertical distance formula:

\(\text{Vertical Distance} = h = \tan\left(AOE\right) * d\)

\(\text{Vertical Distance} = h = tan\left(45^{o}\right) * 10 meters\)

As \(\tan\left(45^{o}\right)\) = 1, we have \(\text{Vertical Distance} = h = 1 * 10 meters\) \(\text{Vertical Distance} = h = 10 meters\)

A person standing on the ground sees another person working at a certain height of 30m at an angle of \(45^{o}\). What is the horizontal distance between both of them?

**Solution:**

**Data Given:**

- AOE=\(45^{o}\)
- h=30m

**Calculations:**

\(tan\left(\theta\right)=\dfrac{Vertical height}{\text{Horizontal distance}}\)

\(tan\left(45^{o}\right)=\dfrac{30}{\text{Horizontal distance}}\)

As \(tan\left(45^{o}\right)=1\),

So we have; \(1=\dfrac{30}{\text{Horizontal distance}}\) or; \(\text{Horizontal distance}=30m\)

The angle of elevation has one horizontal arm and one arm above the horizontal, which can equal to \(90^{o}\) or less.

Here are some uses of the calculator:

- Estimate the height of tall objects, like trees, buildings, hills, etc
- Find tilt angles for satellite dishes that help in receiving maximum signal strength
- Architects and civil engineers can use the tool to ensure ramps or walkways meet specific slope requirements
- Determine the exact camera tilt angle for the desired composition
- Calculate elevation grade to examine the flow of water from higher to lower areas to verify that agricultural land can be irrigated without water pumps
- Knowing the elevation grade is crucial for building safe and effective retaining walls. So that these walls can hold back soil to prevent erosion and landslides, often on slopes

It is the measure of steepness (slope) of a surface for a specific area of land.

- If terrain grade
**< 1**, the slope is gentle - If terrain grade
**> 1**, the slope is relatively steep - If terrain grade
**= 1**, the slope is a 100% rise over a 100% run (for every unit increase in the run, there is an equal unit increase for rise)

There is no direct method of calculating the angle of elevation using trigonometry without additional information (rise and run). Trigonometry requires a relationship between sides of a triangle to solve for angles. While some online tools might be able to help in calculating the angle of elevation indirectly.

**References:**

Wikipedia:** **Elevation

Khan Academy: Intro to radians, Radian angles & quadrants, Tangent identities

Lumen Learning: Trigonometric Functions, Trigonometric Identities, Key Equations

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