Angle of Elevation
The 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. Formally, it is the angle formed between the horizontal line of sight and the line of sight to the object; if the horizontal distance is $d$ and the object's height above eye level is $h$, then $\tan(\text{elevation angle}) = h/d$, 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
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. A surveyor $80$ m from a building who measures an elevation angle of $35^\circ$ finds the building height $= 80 \times \tan(35^\circ) \approx 56$ m above eye level. With two observers $100$ m apart measuring elevation angles of $40^\circ$ and $55^\circ$ to a tower top, the law of sines on the resulting triangle determines the tower's height without direct measurement.
Key Insight
Start by looking straight forward, that is $0^\circ$; the more you tilt up toward the sky, the bigger the angle of elevation gets, and it is always measured from the horizontal, never from the vertical, a common error to avoid. 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.