ACT® · Math: Geometry — Similarity, Right Triangles, and Trigonometry
Angles of Elevation and Depression
An angle of elevation is measured up from horizontal to a higher point; an angle of depression is measured down from horizontal to a lower point. They're always equal between two people looking at each other, and both set up a right triangle with the horizontal as one leg.
Picture a horizontal line straight out from your eye. Elevation is how far you tilt your gaze up from that line to spot something above you; depression is how far you tilt down to spot something below. If you're on the ground looking up at a tower at some angle, the person at the top looking down at you is measuring the angle between their own horizontal line and the line of sight — and that line of sight is the same line either way. The two horizontal lines are parallel, so by alternate interior angles, the elevation angle and the depression angle are equal.
To turn the scene into a triangle, draw the horizontal leg along the ground (or along eye level), the vertical leg as the height difference, and the line of sight as the hypotenuse. The angle sits between the horizontal leg and the hypotenuse. That's what makes tangent the natural tool: tangent equals opposite over adjacent, so tan(angle) = vertical leg / horizontal leg. Once you know which two sides you want and which one you're given, tangent alone solves almost every one of these.
The catch in most setups is eye level: a person's eyes aren't at ground level, so the triangle actually sits on a line y feet up, not on the ground itself. Tangent gives you only the triangle's vertical leg, the height above or below eye level — you still have to add or subtract y to get the true height. This topic sits right next to the two special right triangles and the basic trig ratios, because it's really just those ratios applied to a vertical scene instead of an abstract one.
Key terms
- Angle of elevation
- The angle measured upward from a horizontal line to the line of sight toward a higher object.
- Angle of depression
- The angle measured downward from a horizontal line to the line of sight toward a lower object.
- Alternate interior angles
- Equal angles formed when a line crosses two parallel lines, which is why elevation and depression angles match.
- Tangent ratio
- In a right triangle, tan(angle) = opposite side / adjacent side, the main tool for these problems.
- Eye level
- The height of the observer's eyes above the ground, which must be added or subtracted to find true height.
Watch it explained
- How it worksHow does an elevation problem turn a real scene into a right triangle with the horizontal ground as one leg?
- Worked exampleFind a tree's height when a person 50 ft away sights the top at 38 degrees from an eye level of 5 ft.
- Worked exampleFind how far a boat sits from the base of a 120 ft cliff when the angle of depression to it is 24 degrees.
- Going furtherWhat happens to the answer when the observer's own eye level gets left out of the setup?
Check yourself
End of topic.