Eduqo
Quantitative Aptitude›Ch 4: Algebra, Geometry & Mensuration›Heights and Distances
Mastering 30°, 45°, and 60° angle problems for competitive exams
Always sketch a clear right-angled triangle showing the line of sight, angle of elevation or depression, and the given measurements. Use standard trigonometric ratios or side length proportions for 30-60-90 and 45-45-90 triangles to solve quickly without lengthy calculations.
Quick Memory Hooks
| Angle | Side Ratio (P : B : H) | Key Shortcut |
|---|---|---|
| 30° | 1 : √3 : 2 | Base = √3 × P |
| 45° | 1 : 1 : √2 | Base = Height |
| 60° | √3 : 1 : 2 | Height = √3 × B |
Elevation
(Looking Up)
Looking Up
Observer looks up at the top of a tower.
Depression
(Looking Down)
Looking Down
Person looks down from a cliff to a boat.
Line of Sight
(The Hypotenuse)
The Hypotenuse
The direct path from eye to object.
Shadow
(Ground Distance)
Ground Distance
Length of pole's shadow cast by the sun.
Confusing elevation and depression anglesElevation and depression are equal
Depression from top equals elevation from bottom due to parallel lines.
Ignoring observer's eye heightSubtract observer height from total
If observer height is given, subtract it from total height before calculation.
Interchanging base and perpendicular in 30°Remember tan 30° = 1 / √3
For 30 degrees, the perpendicular is smaller and the base is root 3 times larger.
Q1.The angle of elevation of the top of a tower from a point on the ground 30 m away from its foot is 30°. Find the height of the tower.
Q2.A ladder 15 m long reaches a window 15/2 m high above the ground. What is the angle made by the ladder with the ground?