Angle of Elevation
TrigonometryThe angle of elevation is the angle measured upward from a horizontal line to the line of sight toward an object above.
Formula
\tan(\theta) = \frac{\text{height}}{\text{horizontal distance}}
Definition
The angle of elevation is how far you tilt your eyes upward from looking straight ahead (horizontal) to look at something above you, like the top of a building or a bird in the sky.
Example
You stand $50$ feet from a flagpole and tilt your head up $30^\circ$ to see the top. That $30^\circ$ is the angle of elevation.
Key Insight
Start by looking straight forward (that's $0^\circ$). The more you tilt up toward the sky, the bigger the angle of elevation gets.
Definition
The angle of elevation is the angle formed between the horizontal line of sight and the line of sight to an object above the observer. If the horizontal distance to an object is $d$ and the object's height above eye level is $h$, then $\tan(\text{elevation angle}) = h/d$.
Example
A surveyor $80$ m from a building measures an angle of elevation of $35^\circ$ to the top. Building height $= 80 \times \tan(35^\circ) \approx 80 \times 0.700 = 56$ m (above eye level).
Key Insight
Angles of elevation and depression are always measured from the horizontal, never from the vertical. This is a common error to avoid: the angle is not measured from straight up.
Definition
The angle of elevation $\theta$ satisfies $\tan(\theta) = \text{vertical\_rise} / \text{horizontal\_run}$, precisely the slope of the line of sight. In 3D surveying, two observers at known positions measure angles of elevation to a point, and triangulation via the law of sines determines the point's position.
Example
Two observers A and B are $100$ m apart. A measures an elevation angle of $40^\circ$ and B measures $55^\circ$ to a tower top. Using the law of sines on the resulting triangle determines the tower's height without direct measurement.
Key Insight
The angle of elevation is the complement of the zenith angle (the angle from directly overhead). In astronomy, the altitude of a celestial object equals its angle of elevation above the horizon.