Height and Distance in SSC CGL: The Only Idea You Need
Every height and distance question is a right-angled triangle where you know one side and one angle. You almost always use tan θ = height / base, with θ = 30°, 45° or 60°. SSC CGL Tier 1 usually has 1 question; Tier 2 has 1 to 2, often with two angles.
Key takeaway: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3. Angle of elevation = looking up; angle of depression = looking down, and it equals the angle of elevation from the other point (alternate angles). Two angles A and B seen from points d apart on the same side: h = d·tanA·tanB / (tanB − tanA). Take √3 = 1.732.
Every numerical answer below was computed in Python. Revise the ratios in our trigonometry formulas for SSC CGL first.
Trigonometric Values Table for Height and Distance
| θ | sin θ | cos θ | tan θ | cot θ |
|---|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 | √3 |
| 45° | 1/√2 | 1/√2 | 1 | 1 |
| 60° | √3/2 | 1/2 | √3 | 1/√3 |
Ratio trick: in a 30-60-90 triangle the sides are 1 : √3 : 2 (opposite 30°, opposite 60°, hypotenuse). In a 45-45-90 triangle they are 1 : 1 : √2.
Height and Distance Formulas (Shortcuts)
| Situation | Formula |
|---|---|
| Height h, base d, elevation θ | h = d·tan θ ; d = h·cot θ |
| Ladder / string of length L at angle θ with the ground | height = L·sin θ ; foot distance = L·cos θ |
| Observer of height a, tower at distance d, elevation θ | h = a + d·tan θ |
| Elevation changes from A to B (B > A) as you walk d metres towards the tower | h = d·tanA·tanB / (tanB − tanA) |
| Same: 30° to 60° | h = (√3/2)·d |
| Two objects seen from a height h at depressions A and B (same side) | distance between them = h(cotA − cotB) |
| Two objects on opposite sides at elevations A and B | distance between them = h(cotA + cotB) |
| Flagstaff on a building: elevations A (building top), B (flagstaff top), building height H | distance = H·cotA ; flagstaff = distance·tanB − H |
Solved Height and Distance Questions (SSC CGL Pattern)
- A tower casts a 20 m shadow when the sun's elevation is 60°. Height? h = 20·tan 60° = 20√3 ≈ 34.64 m.
- A 1.5 m tall man stands 30 m from a tower and sees its top at 45°. Height of tower? 1.5 + 30·1 = 31.5 m.
- The elevation of a tower's top changes from 30° to 60° when you walk 40 m towards it. Height? (√3/2)·40 = 20√3 ≈ 34.64 m.
- A 10 m ladder makes 60° with the ground. Height it reaches on the wall? 10·sin 60° = 5√3 ≈ 8.66 m.
- From the top of a 75 m tower, a car is seen at a depression of 30°. Distance of the car from the foot? 75·cot 30° = 75√3 ≈ 129.9 m.
- A kite string of 100 m makes 30° with the ground. Height of the kite? 100·sin 30° = 50 m.
- The shadow of a pole equals its height. Sun's altitude? tan θ = 1, so θ = 45°.
- From the top of a 100 m cliff, two boats on the same side are seen at depressions of 30° and 45°. Distance between them? 100(√3 − 1) = ≈ 73.2 m.
- Two points on opposite sides of a 60 m tower see its top at 60° and 30°. Distance between the points? 60(1/√3 + √3) = 80√3 ≈ 138.56 m.
- From a point, the top of a 20 m building is at 30° and the top of a flagstaff on it is at 45°. Length of flagstaff? Distance = 20√3; flagstaff = 20√3 − 20 = 20(√3 − 1) ≈ 14.64 m.
Height and Distance Tricks to Save Time
- Answer has √3? Most SSC answers come as 10√3, 20√3, 50(√3 + 1) and so on. Keep the √3 until the end, then multiply by 1.732 only if options are decimal.
- 30° to 60° shortcut: height = (√3/2) × distance walked, and the observer's final distance from the tower = half the distance walked.
- Depression = elevation: turn every depression angle into the same elevation angle at the other point.
- Draw it: a 5-second sketch with the right angle marked avoids most sign errors.
Common Mistakes in Height and Distance
- Using sin when the question gives the base (use tan), or tan when it gives the string/ladder length (use sin or cos).
- Forgetting to add the observer's height.
- Taking the depression angle inside the triangle at the top instead of transferring it to the ground point.
- Using tan 60° = 1/√3 (that is tan 30°).
Height and Distance FAQs
What is the angle of elevation?
The angle between the horizontal line from the eye and the line of sight to an object above the eye. The angle of depression is the same idea for an object below the eye.
Which formula is used most in height and distance?
tan θ = height / base. For a ladder or string, use height = length × sin θ.
What is the shortcut when the angle changes from 30° to 60°?
If you walk d metres towards the object and the elevation changes from 30° to 60°, the height is (√3/2) × d.
How many height and distance questions come in SSC CGL?
Usually about 1 in Tier 1 and 1 to 2 in Tier 2, often combined with trigonometric values.
Practice Next
- New SSC CGL tests (Tier 1 level): SSC CGL Trigonometry Mock Test 02.
- Mock tests: SSC CGL Trigonometry Mock Test 01, SSC CGL Quantitative Aptitude Sectional Mock Test 03 and SSC CHSL Quantitative Aptitude Sectional Mock Test 04.
- Topic test: SSC CGL Height and Distance Mock Test 01.
- Related notes: trigonometry formulas for SSC CGL, mensuration formulas for SSC CGL, coordinate geometry formulas for SSC CGL and algebra formulas for SSC CGL.
- Exam details: SSC CGL 2026.
- Browse all study notes.