Height and Distance Formula Guide: Angle of Elevation, Depression and Two-Angle Questions for SSC

Height and distance questions come from one idea: a right-angled triangle and the ratio tan θ = height / distance. This height and distance formula guide for SSC gives the standard values you need, the two-angle formula, and the solved examples on towers, shadows, ladders and depression.

Basic Terms

The angle of elevation of an object from a point is equal to the angle of depression of that point from the object (alternate angles).

Values You Need

Main Formulas

Solved Examples

Example 1. The angle of elevation of the top of a tower from a point 20 m from its base is 45°. Find the height.
h = 20 × tan 45° = 20 m.

Example 2. The angle of elevation is 60° at a distance of 20 m. Find the height.
h = 20 × √3 = 20√3 m.

Example 3. The angle of elevation of the top of a tower is 30°. After walking 20 m towards the tower, it is 60°. Find the height.
h = 20 / (cot 30° − cot 60°) = 20 / (√3 − 1/√3) = 20 / (2/√3) = 10√3 m.

Example 4. From the top of a 100 m tower, the angle of depression of a car is 45°. How far is the car from the foot of the tower?
d = 100 / tan 45° = 100 m.

Example 5. A pole 10 m high casts a shadow of 10√3 m. Find the angle of elevation of the sun.
tan θ = 10 / (10√3) = 1/√3, so θ = 30°.

Example 6. A ladder 10 m long makes an angle of 60° with the ground. Find the height reached and the distance of its foot from the wall.
Height = 10 × sin 60° = 5√3 m. Distance = 10 × cos 60° = 5 m.

Quick Tips

• Draw the figure and mark the right angle.
• Use tan for height and distance questions.
• The angle of elevation equals the angle of depression.
• Remember the values for 30°, 45° and 60°.

Quick Revision Points

• tan θ = height / distance.
• Elevation equals depression (alternate angles).
• Two-angle formula: h = x / (cot α − cot β).
• tan 30° = 1/√3, tan 45° = 1 and tan 60° = √3.
• A ladder gives height = L sin θ.

Practice Questions With Answers

1. The distance is 20 m and the angle of elevation is 45°. Find the height.
   Answer: 20 m
2. The distance is 10 m and the angle is 60°. Find the height.
   Answer: 10√3 m
3. The distance is 30 m and the angle is 30°. Find the height.
   Answer: 10√3 m
4. A tower of 50 m has a shadow of 50 m. Find the angle of elevation of the sun.
   Answer: 45°
5. A ladder 10 m long makes 30° with the ground. Find the height reached.
   Answer: 5 m
6. From a 100 m tower, the angle of depression of a car is 45°. Find the distance of the car.
   Answer: 100 m
7. A pole is 10√3 m high and the angle of elevation is 60°. Find the distance.
   Answer: 10 m
8. What is the value of tan 30°?
   Answer: 1/√3
9. Is the angle of elevation equal to the angle of depression?
   Answer: Yes (alternate angles)
10. The angles are 30° and 60° after walking 20 m nearer. Find the height.
    Answer: 10√3 m

How to Practise This Chapter

Write each formula on a card, with a solved example on the back, and revise the cards every day. Then test yourself with our practice papers and the previous year papers page. See all guides on the Formula Guide page, and also read the Mathematics notes and Quantitative Aptitude notes. The previous guide is Surds, Indices and Simplification and the next guide is Triangle Geometry Theorems. Also read Trigonometry Tricks.

Frequently Asked Questions

What is the angle of elevation?

It is the angle above the horizontal when you look up at an object.

What is the angle of depression?

It is the angle below the horizontal when you look down at an object.

Which ratio is used in height and distance?

Usually tan θ = height / distance.

What is the two-angle formula?

h = x / (cot α − cot β), where x is the distance moved.

What is the value of tan 60 degrees?

It is √3.

What is the value of tan 30 degrees?

It is 1/√3.

How do I find the height of a ladder's top?

Height = ladder length × sin of the angle with the ground.

Is the angle of elevation equal to the angle of depression?

Yes, for the two points, as they are alternate angles.

Is this topic asked in SSC exams?

Yes. Height and distance is asked in SSC CGL maths.

How much time should I take?

About 40 to 60 seconds.

What are common mistakes?

Using sin or cos in place of tan and not drawing the figure.

How can I learn the formulas?

Draw one figure for each case and write the formula beside it.