Heights and Distances
🔒 Log in to trackApplied trigonometry: angles of elevation and depression, the 30-45-60 rules, two-angle configurations, moving observers and compound figures. Every question reduces to one right triangle and one tangent — learn the three standard-angle multipliers and the cot-difference formula and this becomes near-guaranteed marks.
One page per subtopic: detailed notes, every question type, formulas, tricks and practice sets.
Every formula on one printable page, grouped by subtopic.
5 exam-level questions worked step by step.
67 questions — untimed practice or a timed test with analysis.
Track record in the exam
Questions per shift in recent SSC CGL papers.
Test difficulty mix (67 questions)
Question patterns exams keep repeating
Taken from previous-year papers. If a pattern is marked "very common", expect to see it in your exam.
Single angle, one side known
very commonOne angle (30°, 45° or 60°) is given with the height or the distance.
45°: distance = height.
30°: distance = height × √3. 60°: distance = height ÷ √3.
The angle of elevation of the top of a 20 m tower is 30°. How far is the point from the base?
30° rule: distance = height × √3
Distance = 20 × √3
So the point is 20√3 m away
Looking down (angle of depression)
very commonA lighthouse, cliff or plane looks down at a ship, car or building.
Depression from the top = elevation from the object.
Then use the same 45° / 30° / 60° rules.
From a 100 m cliff, a boat is seen at a depression of 30°. How far is the boat from the cliff base?
Depression 30° from top = elevation 30° from the boat
30° rule: distance = height × √3
Distance = 100 × √3 = 100√3 m
Two angles from two spots
very commonYou walk closer (or two people stand apart) and the angle changes, e.g. 30° to 60°.
Call the height h and write each distance in terms of h.
The difference of the two distances is the distance walked.
Walking 40 m towards a tower, the elevation changes from 30° to 60°. Find the height.
Far spot: h × √3. Near spot: h ÷ √3
Gap = h × √3 − h ÷ √3 = 2h ÷ √3
2h ÷ √3 = 40, so h = 40 × √3 ÷ 2
Height = 20√3 m
Moving car or boat
commonA car or boat moves at a steady speed and the angle changes after some time.
Distance moved = speed × time.
Then treat it like a two-angle problem.
A car moves at 6 m/s. The angle of depression of it from a tower changes from 30° to 60° in 6 s. Find the tower's height.
Distance moved = 6 × 6 = 36 m
Far − near = h × √3 − h ÷ √3 = 2h ÷ √3
2h ÷ √3 = 36, so h = 36 × √3 ÷ 2
Height = 18√3 m
Object on top of another
commonA statue on a pedestal, or a flagstaff on a tower, seen at two angles from one point.
Find the full height and the lower part separately.
Subtract to get the top part.
A statue stands on a 30 m pedestal. From a point, the pedestal top is at 45° and the statue top at 60°. Find the statue's height.
45° on the pedestal top → distance = 30 m
Full height at 60° = 30 × √3
Statue = 30√3 − 30 = 30(√3 − 1) m
Shadow problems
commonA shadow length is given for one object, and you need the height of another.
At the same time, height and shadow grow in the same ratio.
Set up a simple proportion.
A 1.5 m stick casts a 2.5 m shadow. At the same time a tower casts a 75 m shadow. Find the tower's height.
Shadow grows from 2.5 to 75, which is 30 times
Height grows 30 times too
1.5 × 30 = 45 m
Tree that breaks
occasionalA tree breaks and its top touches the ground at some distance, making an angle.
Standing part + broken part = original height.
Use the 30-60-90 triangle to get both parts.
A tree breaks and its top touches the ground 15 m from the base, making 30° with the ground. Find the original height.
Standing part is opposite 30°, so it is half the broken part
Base 15 = standing × √3, so standing = 15 ÷ √3 = 5√3
Broken part = 2 × 5√3 = 10√3
Total = 5√3 + 10√3 = 15√3 m
Ladder, rope or wire
occasionalThe slanting length (ladder, wire, thread) is given with its angle.
Use the 30-60-90 triangle: sides are in the ratio 1 : √3 : 2.
The slanting length is the "2".
A 60 m cable makes 60° with the ground. How tall is the pole?
Sides are in ratio 1 : √3 : 2
Cable (2 parts) = 60, so 1 part = 30
Pole is opposite 60° = √3 parts = 30 × √3 = 30√3 m