Heights and Distances

Also called

angle of elevation · angle of depression · line of sight · clinometer · theodolite · trigonometric ratios · tan θ · tangent ratio · height of a tower · shadow of a tower · lighthouse and ship · two angles one unknown · applications of trigonometry · horizontal level · ऊँचाई और दूरी · उन्नयन कोण · अवनमन कोण

Topics

• The line of sight joins the eye to the object looked at; the angle of elevation is its angle above the horizontal.
• The angle of depression is the angle of the line of sight below the horizontal when the object is lower than the eye.
• A clinometer measures these angles: a protractor with a sight tube, read where a hanging plumb line crosses its scale.
• In the right triangle of eye, tower foot and tower top, tan θ = height above eye level ÷ horizontal distance.
• At 45° the height above the eye equals the distance; at 60° it is √3 times it: closer means a steeper angle.
• Shadow method: the sun’s rays are parallel, so a stick and a tower have shadows in the same ratio as their heights.
• The angle of depression from a lighthouse to a boat equals the boat’s angle of elevation to the lamp (alternate angles).
• The boat sails in and the angle grows from 30° to 60°; both triangles share the lighthouse height, giving the distance.
• Heights of towers, trees and hills, and distances across rivers, are found this way without measuring them directly.

About this model

A GLB file of about 293 KB, shown in Lably's 3D viewer and, on a phone, placed in your room with AR. Credit: Model, science and animation: Lably (built procedurally with three.js; no downloaded meshes). | CC BY 4.0. Original geometry and animation made for Lably.

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