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Height and Distance — Questions and Answers

Height and Distance applies basic trigonometry to a right triangle formed by a vertical object, its horizontal distance and the line of sight. You only ever need three ratios and three standard angles. Draw the triangle, mark the angle of elevation or depression at the correct vertex, and the question becomes a one-line tan calculation.

21 solved questionsArithmetic AptitudeFree · no signup

Height and Distance— Concepts, Formulas & Shortcuts

  • Angle of ELEVATION is measured upward from the horizontal at the observer; angle of DEPRESSION is measured downward.
  • tan θ = height / base is the workhorse; use sin and cos only when the hypotenuse (a ladder, a rope) is involved.
  • Standard values: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3. Take √3 ≈ 1.732.
  • Angle of depression from the top equals the angle of elevation from the bottom (alternate angles).
  • If the shadow equals the height, the sun's elevation is 45°; if the shadow is √3 times the height, it is 30°.
  • Always draw the figure — mis-placing the angle is the single biggest error in this topic.

Height and Distance Practice Questions with Answers

Attempt each question first, then open the explanation. All 21 questions below are free to read and require no signup.

  1. Q1.The angle of elevation of the top of a tower from a point 30√3 m away from its foot is 30°. The height of the tower is:

    Moderate
    • A30 m
    • B45 m
    • C60 m
    • D90 m
    +Show Answer & Explanation

    Answer: A. 30 m

    Explanation: tan 30° = h/(30√3) → h = 30√3 × 1/√3 = 30 m.

    Height and Distance question 1 of 21

  2. Q2.A ladder 10 m long leans against a wall making an angle of 60° with the ground. The height reached on the wall is:

    Moderate
    • A5 m
    • B5√3 m
    • C10√3 m
    • D20/√3 m
    +Show Answer & Explanation

    Answer: B. 5√3 m

    Explanation: The ladder is the hypotenuse, so height = 10 sin 60° = 10 × √3/2 = 5√3 ≈ 8.66 m.

    Height and Distance question 2 of 21

  3. Q3.The angle of elevation of the top of a pole from a point 100 m away is 45°. The height of the pole is:

    Easy
    • A50 m
    • B100 m
    • C100√3 m
    • D141 m
    +Show Answer & Explanation

    Answer: B. 100 m

    Explanation: tan 45° = 1, so height = base = 100 m.

    Height and Distance question 3 of 21

  4. Q4.The angle of elevation of the top of a 100 m tower is 30°. The distance of the observer from the foot of the tower is:

    Moderate
    • A100 m
    • B100/√3 m
    • C173.2 m
    • D200 m
    +Show Answer & Explanation

    Answer: C. 173.2 m

    Explanation: tan 30° = 100/d → d = 100√3 ≈ 173.2 m.

    Height and Distance question 4 of 21

  5. Q5.If the shadow of a pole is √3 times its height, the angle of elevation of the sun is:

    Moderate
    • A30°
    • B45°
    • C60°
    • D75°
    +Show Answer & Explanation

    Answer: A. 30°

    Explanation: tan θ = h/(√3 h) = 1/√3, so θ = 30°.

    Height and Distance question 5 of 21

  6. Q6.When the length of a vertical pole's shadow equals its height, the sun's angle of elevation is:

    Easy
    • A30°
    • B45°
    • C60°
    • D90°
    +Show Answer & Explanation

    Answer: B. 45°

    Explanation: tan θ = h/h = 1, so θ = 45°.

    Height and Distance question 6 of 21

  7. Q7.From the top of a 60 m high cliff, the angle of depression of a boat is 30°. The distance of the boat from the foot of the cliff is:

    Moderate
    • A60 m
    • B60/√3 m
    • C103.9 m
    • D120 m
    +Show Answer & Explanation

    Answer: C. 103.9 m

    Explanation: The angle of depression equals the angle of elevation from the boat: tan 30° = 60/d → d = 60√3 ≈ 103.9 m.

    Height and Distance question 7 of 21

  8. Q8.The angle of elevation of the top of a tower from a point 50 m away is 45°. The height of the tower is:

    Easy
    • A25 m
    • B50 m
    • C50√3 m
    • D100 m
    +Show Answer & Explanation

    Answer: B. 50 m

    Explanation: tan 45° = 1, so height equals the horizontal distance: 50 m.

    Height and Distance question 8 of 21

  9. Q9.A tower is 30 m high. The angle of elevation of its top from a point on the ground is 60°. The distance of the point from the base is:

    Moderate
    • A10√3 m
    • B15 m
    • C30√3 m
    • D60 m
    +Show Answer & Explanation

    Answer: A. 10√3 m

    Explanation: tan 60° = 30/d → d = 30/√3 = 10√3 ≈ 17.32 m.

    Height and Distance question 9 of 21

  10. Q10.A kite is flying at a height of 60 m. The string makes an angle of 30° with the ground. The length of the string is:

    Moderate
    • A60 m
    • B90 m
    • C120 m
    • D60√3 m
    +Show Answer & Explanation

    Answer: C. 120 m

    Explanation: sin 30° = 60/L → L = 60/0.5 = 120 m.

    Height and Distance question 10 of 21

  11. Q11.The angle of elevation of the sun when the length of a pole's shadow is 1/√3 times its height is:

    Moderate
    • A30°
    • B45°
    • C60°
    • D75°
    +Show Answer & Explanation

    Answer: C. 60°

    Explanation: tan θ = h/(h/√3) = √3, so θ = 60°.

    Height and Distance question 11 of 21

  12. Q12.A ladder leans against a wall reaching a height of 12 m, with its foot 5 m from the wall. The length of the ladder is:

    Easy
    • A11 m
    • B13 m
    • C15 m
    • D17 m
    +Show Answer & Explanation

    Answer: B. 13 m

    Explanation: √(12² + 5²) = √169 = 13 m.

    Height and Distance question 12 of 21

  13. Q13.From the top of a lighthouse 75 m high, the angle of depression of a ship is 45°. The ship's distance from the base is:

    Easy
    • A37.5 m
    • B75 m
    • C75√3 m
    • D150 m
    +Show Answer & Explanation

    Answer: B. 75 m

    Explanation: With a 45° depression, tan 45° = 1, so the horizontal distance equals the height: 75 m.

    Height and Distance question 13 of 21

  14. Q14.Two poles of equal height stand on either side of a road. From a point between them the angles of elevation of their tops are 60° and 30°. The distances from the point to the nearer and the farther pole are in the ratio:

    Difficult
    • A1 : 2
    • B1 : 3
    • C2 : 3
    • D1 : √3
    +Show Answer & Explanation

    Answer: B. 1 : 3

    Explanation: For equal heights h, distance = h/tan θ. Nearer pole (60°): h/√3. Farther pole (30°): h√3. Ratio = (h/√3) : (h√3) = 1 : 3.

    Height and Distance question 14 of 21

  15. Q15.An observer 1.6 m tall is 20√3 m from a tower. The angle of elevation of the tower's top is 30°. The height of the tower is:

    Difficult
    • A20 m
    • B21.6 m
    • C23.2 m
    • D24 m
    +Show Answer & Explanation

    Answer: B. 21.6 m

    Explanation: tan 30° = x/(20√3) → x = 20 m above eye level. Adding the observer's height: 20 + 1.6 = 21.6 m.

    Height and Distance question 15 of 21

  16. Q16.The angle of elevation of the top of a 20 m tower from a point on the ground is 45°. The distance of the point from the base is:

    Easy
    • A10 m
    • B20 m
    • C20√3 m
    • D40 m
    +Show Answer & Explanation

    Answer: B. 20 m

    Explanation: tan 45° = 1, so the distance equals the height: 20 m.

    Height and Distance question 16 of 21

  17. Q17.From a point 10 m from the foot of a tower, the angle of elevation of its top is 60°. The height of the tower is:

    Moderate
    • A10 m
    • B10√3 m
    • C20 m
    • D20√3 m
    +Show Answer & Explanation

    Answer: B. 10√3 m

    Explanation: tan 60° = h/10 → h = 10√3 ≈ 17.32 m.

    Height and Distance question 17 of 21

  18. Q18.A ladder 26 m long has its foot 10 m from a wall. The height it reaches on the wall is:

    Moderate
    • A16 m
    • B20 m
    • C24 m
    • D25 m
    +Show Answer & Explanation

    Answer: C. 24 m

    Explanation: √(26² − 10²) = √(676 − 100) = √576 = 24 m.

    Height and Distance question 18 of 21

  19. Q19.A 15 m high pole casts a shadow such that the sun's elevation is 30°. The length of the shadow is:

    Moderate
    • A15 m
    • B15√3 m
    • C30 m
    • D5√3 m
    +Show Answer & Explanation

    Answer: B. 15√3 m

    Explanation: tan 30° = 15/s → s = 15√3 ≈ 25.98 m.

    Height and Distance question 19 of 21

  20. Q20.From the top of a 30 m tower, the angle of depression of a car is 60°. The car's distance from the base is:

    Moderate
    • A10√3 m
    • B15 m
    • C30 m
    • D30√3 m
    +Show Answer & Explanation

    Answer: A. 10√3 m

    Explanation: tan 60° = 30/d → d = 30/√3 = 10√3 ≈ 17.32 m.

    Height and Distance question 20 of 21

  21. Q21.If the angle of elevation of the sun decreases, the length of a pole's shadow:

    Moderate
    • AIncreases
    • BDecreases
    • CStays the same
    • DBecomes zero
    +Show Answer & Explanation

    Answer: A. Increases

    Explanation: Shadow length = h/tan θ. As θ falls, tan θ falls and the shadow lengthens.

    Height and Distance question 21 of 21

Height and Distance — Frequently Asked Questions

What is the difference between angle of elevation and angle of depression?+

Elevation is the angle you look UP through from the horizontal to see an object above you; depression is the angle you look DOWN through to see an object below you. They are numerically equal for the same pair of points, since the two horizontals are parallel.

How do I know whether to use sin, cos or tan?+

If the question gives or asks for the slant length (a ladder, a wire, a rope), use sin or cos. If it involves only the vertical height and the horizontal ground distance, use tan.

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