Angle of Elevation and Depression

Pre-Calculus · Axiom Academy

Angle of Elevation & Depression One right triangle — a horizontal line, a line of sight, and the angle between them — turns a measured angle into a height or a distance. A lighthouse keeper, 120 ft up, spots a ship and measures the angle down to it. From just that angle she finds how far away it is — and the same trick, looked at upward , finds heights you can't reach. Look down: how far is the ship? Set the angle of depression, hit Scan, and watch the keeper's line of sight drop from the horizon to the ship. The shallower the angle, the farther the ship — that's d = height ÷ tan(θ) , live. Now stand 80 ft from a tower and tilt your gaze up to the top — that's the angle of elevation . Same triangle, flipped: with the distance fixed, the height is h = distance × tan(θ) . (Look back down from the top and it's the very same angle — elevation = depression.) Walk closer — how much steeper? The tower's height never changes — but where you stand does. You're 300 ft from a 200 ft tower; step toward it and watch the angle of elevation climb. Closer ground means a steeper look up: θ = the angle whose tangent is 200 ÷ distance . One triangle, two views: look down to find a distance, look up to find a height — and the angle is the same either way. Anywhere you can measure an angle off the horizontal — a ship from a lighthouse, a plane on its glide slope, a tree from across the field, a building from the curb — tan(θ) trades that angle for the side you can't reach.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.