Surveyors, pilots and builders measure angles from the horizontal. Looking up at something, the angle between the horizontal and your line of sight is the angle of elevation. Looking down, it is the angle of depression. Each makes a right triangle you can solve with Finding Angles with Inverse Trig and Solving Right Triangles.
Why it works
Picture a person on a cliff looking down at a boat. The horizontal line at the cliff top and the ground (or water) are parallel, and the line of sight is a transversal. The angle of depression at the top and the angle of elevation at the boat are alternate interior angles, so they are equal. That lets you move the angle into the right triangle.
Key idea
Angles of elevation and depression are measured from a horizontal line, never from a vertical one. The angle of depression from $A$ to $B$ equals the angle of elevation from $B$ to $A$.
Worked example: A height
From $40$ m away on level ground, the angle of elevation to the top of a tower is $55^\circ$. How tall is the tower?
- The height is opposite the angle; the $40$ m distance is adjacent: $\tan 55^\circ = \frac{h}{40}$.
- $h = 40 \tan 55^\circ \approx 40(1.428) \approx 57.1$ m.
Answer: About $57.1$ m
Worked example: A distance from a depression angle
From the top of a $90$ ft lighthouse, the angle of depression to a boat is $12^\circ$. How far is the boat from the base?
- Move the angle to the boat: the angle of elevation there is $12^\circ$.
- $\tan 12^\circ = \frac{90}{x}$, so $x = \frac{90}{\tan 12^\circ} \approx \frac{90}{0.2126} \approx 423$ ft.
Answer: About $423$ ft
Worked example: Finding the angle
A $15$ ft ladder reaches $13$ ft up a wall. What angle does it make with the ground?
- The ladder is the hypotenuse and the height is opposite the angle at the ground: $\sin \theta = \frac{13}{15}$.
- $\theta = \sin^{-1}\left(\frac{13}{15}\right) \approx 60.1^\circ$.
Answer: About $60^\circ$
Common mistake
Measuring the angle of depression from the vertical (the side of the lighthouse) instead of the horizontal. That angle is the complement, $90^\circ$ minus the right one, and it gives very wrong distances. Check: draw the horizontal line at the observer's eye first, then mark the angle down from it.