Heights and Distances: Trigonometry Problems Solved

Heights and distances is the practical chapter of trigonometry: it uses angles and one measured distance to find things you cannot measure directly, like the height of a tower or the width of a river. Stand at a known distance, measure the angle to the top, and let the tangent function do the rest.

This matters because direct measurement is often impossible. You cannot climb every tower or swim across every river with a tape. A small angle-measuring instrument and one ground measurement are enough, which is why surveyors have relied on these methods for centuries.

Key Concepts

• Angle of elevation: the angle between the horizontal line from the observer's eye and the line of sight to an object above eye level.
• Line of sight: the straight line from the observer's eye to the object being observed.
• The right triangle: the tower (vertical), the ground distance (horizontal), and the line of sight form a right triangle. The tangent of the angle at the observer equals opposite over adjacent — height over ground distance.
• Two-point observation: when the distance to the object is unknown, observe from two points a known distance d apart along the same line and measure two angles; the height follows from both measurements combined.

Formulas

• Single observation: tan θ = height / distance, so height = distance × tan θ and distance = height / tan θ. θ is in degrees; height and distance share one length unit (metres).
• Two-point observation: if angles α (nearer, larger) and β are measured from two points d metres apart, height h = d / (cot β − cot α).
• Assumptions: the ground is level, the object is vertical, and both observations are taken from the same eye level along a straight line toward the object.

Worked Example

A surveyor stands 50 m from the base of a tower and measures the angle of elevation to its top as 30°. Find the tower's height.

1. Draw the right triangle: horizontal leg 50 m, vertical leg h, angle at the observer 30°.
2. tan 30° = h / 50, so h = 50 × tan 30°.
3. tan 30° ≈ 0.5774, so h ≈ 50 × 0.5774 = 28.87.
4. The tower is about 28.9 m tall.

Common Misconceptions

• Wrong angle: the formula needs the angle at the observer between the horizontal and the line of sight. Measuring from the tower side or from the vertical gives the complementary angle and a wrong answer.
• Skipping the diagram: most errors come from mixing up opposite and adjacent. Always sketch the triangle and label θ first.
• Forgetting eye level: instruments measure from the observer's eye, not the ground. In precise work, add the eye height to the computed value.

Real-World Applications

• Surveying: measuring building heights, hill elevations, and river widths without crossing or climbing.
• Aviation: pilots and air-traffic controllers use elevation and depression angles for descent paths and runway approaches.
• Astronomy: early astronomers estimated distances to the Moon and planets using parallax, the same two-point idea.

Heights and distances turns one measured angle and one measured length into heights you cannot reach. Set up the right triangle with the angle at the observer, apply height = distance × tan θ, and for two observation points combine both angles with the known gap between them.