Quantitative Aptitude

Heights and Distances: Practice Questions with Answers

13 practice questions on Heights and Distances, with answers; most also have an explanation or a worked solution. Try each one before you open its answer.

Read the lesson on this topic →

1. Q.1

   A flagpole stands on top of a 20 m tall tower. From a point on the ground, the angle of elevation of the bottom of the flagpole is 30°, and the angle of elevation of the top of the flagpole is 45°. Find the length of the flagpole.

   Show answer

   Answer: 20(√3 − 1) m ≈ 14.64 m

   Solution:

   1. Let the horizontal distance from the point to the tower be d. From the bottom of the flagpole (top of the tower, height 20): tan 30° = 20/d, so d = 20√3.
   2. From the top of the flagpole (height 20 + L, where L is the flagpole's length): tan 45° = (20 + L)/d, and since tan 45° = 1, this gives 20 + L = d = 20√3.
   3. L = 20√3 − 20 = 20(√3 − 1) ≈ 14.64 m.

   Report this question

2. Q.2

   From a point on the ground, the angle of elevation of the top of a tower is 30°. After walking 20 m towards the tower, the angle becomes 60°. Find the height of the tower.

   Show answer

   Answer: 10√3 m ≈ 17.32 m

   Solution:

   1. Let height = h, and let the original distance be d. From the first position: tan 30° = h/d, so d = h√3.
   2. From the closer position: tan 60° = h/(d − 20), so d − 20 = h/√3.
   3. Substitute: h√3 − 20 = h/√3. Multiply through by √3: 3h − 20√3 = h, so 2h = 20√3, giving h = 10√3 ≈ 17.32 m.

   Report this question

3. Q.3

   A pole casts a shadow of length 10√3 m when the sun’s angle of elevation is 30°. Find the height of the pole.

   Show answer

   Answer: 10 m

   Solution:

   1. tan(30°) = height / shadow length, so height = shadow length × tan(30°).
   2. tan(30°) = 1/√3, so height = 10√3 × (1/√3) = 10 m.

   Report this question

4. Q.4

   From the top of a cliff 100 m high, the angle of depression of a boat is 30°. Find the distance of the boat from the base of the cliff.

   Show answer

   Answer: 100√3 m ≈ 173.2 m

   Solution:

   1. The angle of depression from the top equals the angle of elevation from the boat (alternate angles), so tan(30°) = height / distance.
   2. tan(30°) = 1/√3, so 1/√3 = 100 / distance, giving distance = 100√3 ≈ 173.2 m.

   Report this question

5. Q.5

   A ladder leans against a wall, making an angle of 60° with the ground. If the foot of the ladder is 5 m from the wall, find the length of the ladder.

   Show answer

   Answer: 10 m

   Solution:

   1. The ladder is the hypotenuse, and 5 m is the side adjacent to the 60° angle: cos(60°) = adjacent / hypotenuse.
   2. cos(60°) = 1/2, so hypotenuse = 5 / (1/2) = 10 m.

   Report this question

6. Q.6

   The angle of elevation of the top of a tower from a point 50 m away from its base is 45°. Find the height of the tower.

   Show answer

   Answer: 50 m

   Solution:

   1. tan(45°) = height / distance, so height = distance × tan(45°).
   2. tan(45°) = 1, so height = 50 × 1 = 50 m.

   Report this question

7. Q.7

   A flagpole stands on top of a 10 m tall tower. The angle of elevation of the flagpole's bottom is 30°, and of its top is 60°, from the same point. Find the flagpole's length.

   Show answer

   Answer: 20 m. d = 10√3 from the 30° angle; 10 + L = d√3 = 30, so L = 20.

   Report this question

8. Q.8

   The angle of elevation of the top of a building from a point 40 m away from its base is 45°. Find the height of the building.

   Show answer

   Answer: 40 m

   Report this question

9. Q.9

   From the top of a tower 60 m high, the angle of depression of a car is 45°. Find the distance of the car from the base of the tower.

   Show answer

   Answer: 60 m. tan 45° = 60/distance = 1.

   Report this question

10. Q.10

    A tower casts a shadow of length 20√3 m when the sun’s angle of elevation is 30°. Find the height of the tower.

    Show answer

    Answer: 20 m

    Report this question

11. Q.11

    A ladder leans against a wall, making an angle of 60° with the ground. If the foot of the ladder is 6 m from the wall, find the length of the ladder.

    Show answer

    Answer: 12 m

    Report this question

12. Q.12

    From a point on the ground, the angle of elevation of the top of a tower is 30°. After walking 40 m towards the tower, the angle becomes 60°. Find the height of the tower.

    Show answer

    Answer: 20√3 m ≈ 34.64 m. Same method as the worked example, with 40 m instead of 20 m.

    Report this question

13. Q.13

    A wire of length 20 m is attached from the top of a pole to a point on the ground, making an angle of 30° with the ground. Find the height of the pole.

    Show answer

    Answer: 10 m. sin 30° = height/20, so height = 20 × 1/2.

    Report this question

Practise this topic →