Heights and Distances Notes for SSC Exams

Heights and Distances is an application of trigonometry and one or two questions come from it in each SSC CGL Tier 1 paper. The questions give an angle of elevation or depression and ask for a height or a distance. These heights and distances notes for SSC explain the idea, the formulas and the common question types with solved examples.

Angle of Elevation and Depression

• Angle of elevation: the angle between the horizontal line and the line of sight when we look up at an object.
• Angle of depression: the angle between the horizontal line and the line of sight when we look down at an object.
• The angle of elevation from a point A to B equals the angle of depression from B to A.
• The horizontal line and the vertical line (a tower) are perpendicular.

Basic Formulas

Remember tan 30 = 1/root 3, tan 45 = 1 and tan 60 = root 3. With 45 degrees, height = distance.

Two-Angle Problems

When a person moves closer to a tower and the angle of elevation changes, use the height h to write both distances: d1 = h / tan(theta1) and d2 = h / tan(theta2). The distance moved is d1 - d2. For example, if the angle changes from 30 to 60 degrees, d1 = h root 3 and d2 = h / root 3, so the distance moved = h (root 3 - 1/root 3) = 2h / root 3, which gives h = (root 3 / 2) x distance moved. Solve the equation for h in the same way for other pairs of angles.

Steps to Solve Any Question

1. Draw a clear figure with the horizontal and the vertical.
2. Mark the given angles and distances.
3. Write the trigonometric ratio that connects the known and the unknown sides.
4. Solve for the unknown and check whether it is reasonable.

Solved Examples

Example 1: A tower is 30 m high and its shadow is 30 m long. Find the angle of elevation of the sun.
Solution: tan theta = 30/30 = 1, so theta = 45 degrees.

Example 2: The angle of elevation of the top of a tower from a point 20 m from its base is 60 degrees. Find the height of the tower.
Solution: h = 20 x tan 60 = 20 root 3 m (about 34.6 m).

Example 3: The angle of elevation of the top of a tower is 30 degrees from a point 50 m from its foot. Find the height.
Solution: h = 50 x tan 30 = 50 / root 3 = about 28.9 m.

Example 4: A 10 m ladder reaches a window 8 m above the ground. Find the distance of its foot from the wall.
Solution: Distance = square root of (100 - 64) = 6 m.

Example 5: The angle of depression of a car from the top of a 100 m tower is 45 degrees. How far is the car from the foot of the tower?
Solution: tan 45 = 100 / d, so d = 100 m.

Example 6: A man 1.5 m tall stands 20 m from a tower. The angle of elevation of the top of the tower is 45 degrees. Find the height of the tower.
Solution: Height above the eye = 20 x tan 45 = 20 m. Total height = 20 + 1.5 = 21.5 m.

Quick Tips to Solve Faster

• Draw the figure first; most mistakes happen when the figure is wrong.
• At 45 degrees, height and distance are equal.
• Add the height of the observer to the answer when the eye level is given.
• The angle of elevation and the angle of depression between two points are equal.
• Use the Pythagorean triplets for ladder questions.

Common Mistakes to Avoid

• Using sin or cos instead of tan when height and distance are involved.
• Forgetting to add the observer's height.
• Marking the angle of depression from the vertical instead of the horizontal.
• Mixing up 30 and 60 degrees in the tan values.
• Not simplifying root 3 properly at the end.

Quick Revision Points

• tan theta = height / distance.
• tan 30 = 1/root 3, tan 45 = 1 and tan 60 = root 3.
• Angle of elevation = angle of depression between two points.
• Ladder: foot = l cos theta and height = l sin theta.
• Observer height is added to the calculated height.

Practice Questions With Answers

1. The shadow of a 15 m pole is 15 root 3 m long. Find the angle of elevation of the sun.
   Answer: 30 degrees (tan theta = 1/root 3).
2. The angle of elevation of the top of a tower is 45 degrees from a point 40 m from its base. Find the height.
   Answer: 40 m (h = d).
3. The angle of elevation of the top of a 50 m tower from a point is 30 degrees. Find the distance of the point from the base.
   Answer: 50 root 3 m (50 / tan 30).
4. A 13 m ladder reaches 12 m on a wall. Find the distance of its foot from the wall.
   Answer: 5 m (Triplet 5-12-13).
5. The angle of elevation of a pole top from a point 10 m away is 60 degrees. Find the height.
   Answer: 10 root 3 m (10 x tan 60).
6. The angle of depression of a boat from a 60 m tower is 30 degrees. Find the horizontal distance of the boat from the tower.
   Answer: 60 root 3 m (60 / tan 30).
7. A kite flies at the end of a 50 m string making an angle of 30 degrees with the ground. Find its height.
   Answer: 25 m (50 x sin 30).
8. At what angle of elevation of the sun is the shadow of a pole equal to its height?
   Answer: 45 degrees (tan theta = 1).
9. The angle of elevation of a tower of height 20 root 3 m is 60 degrees. Find the distance of the point from the base.
   Answer: 20 m (20 root 3 / root 3).
10. A man 1.5 m tall sees the top of a tower at an angle of 45 degrees from a distance of 20 m. Find the height of the tower.
    Answer: 21.5 m (20 + 1.5).

How to Practise This Chapter

1. Learn the formulas in the table until you can write them without looking.
2. Solve the solved examples again on your own, without reading the solution.
3. Attempt the practice questions above with a timer of 10 minutes.
4. Solve the matching SSC trigonometry practice paper to test yourself in exam style.
5. Check the previous year papers to see how SSC frames this topic.

Related Notes and Practice Papers

• Trigonometry Notes
• Triangle Geometry Notes
• Circle Geometry Notes
• Square Root and Cube Root Notes
• SSC Trigonometry Practice Paper

Explore More SSC Subjects

SSC exams test many subjects, so connect this topic with the related subjects below:

• Reasoning Notes
• Statistics Notes
• Indian Economy Notes
• English Notes
• General Awareness Notes
• All SSC subjects and notes

Frequently Asked Questions

What is the angle of elevation?

It is the angle between the horizontal and the line of sight when we look up at an object.

What is the angle of depression?

It is the angle between the horizontal and the line of sight when we look down at an object.

Which ratio is used in heights and distances?

The tan ratio is used most often: tan theta = height / distance.

What is the height when the angle is 45 degrees?

The height is equal to the distance because tan 45 = 1.

How many questions come from heights and distances in SSC exams?

One or two questions come in SSC CGL Tier 1, and more in Tier 2.

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

Yes, between two points the angle of elevation from the lower point equals the angle of depression from the upper point.

Do I need to add the height of the observer?

Yes, when the height of the person is given, add it to the calculated height above the eye level.

How should I practise heights and distances?

Learn the tan values, draw the figure for every question and attempt the trigonometry practice paper.