Heights and Distances

Heights and distances is one of the most practical branches of trigonometry, allowing us to calculate measurements that would be physically impossible or impractical to measure directly. By combining known distances or angles with trigonometric ratios — sine, cosine, and tangent — surveyors, architects, engineers, and navigators can determine the height of a mountain, the width of a river, or the distance to a ship at sea without ever leaving the ground. The central tools are the angle of elevation and the angle of depression. The angle of elevation is the angle measured upward from the horizontal line of sight to an object above the observer, while the angle of depression is the angle measured downward from the horizontal to an object below the observer. Both angles are always measured from the horizontal, never from the vertical, and this distinction is critical for setting up correct trigonometric equations. These problems reduce to solving right triangles. Once a right triangle is identified in the geometric situation, the known side and angle information is substituted into a trigonometric ratio to find the unknown. The tangent ratio is most frequently used because it connects the opposite side (often the height) to the adjacent side (often the horizontal distance) without involving the hypotenuse, which is rarely what we need to find directly.

Key takeaways

Draw a clear diagram
Identify the right triangle
Select the correct trigonometric ratio
Solve algebraically, then compute
Two-position problems

Draw a clear diagram

Sketch the physical situation with a horizontal reference line at the observer's eye level. Mark the object whose height or distance is unknown. Label all given measurements — distances along the ground, angles of elevation or depression, and any known heights. A well-drawn diagram almost always reveals which right triangle contains the unknown.

Identify the right triangle

The right angle is always at the base of the vertical height. In a typical 'find the height of a tower' problem, the triangle has vertices at: (A) the observer's position on the ground, (B) the base of the tower, and (C) the top of the tower. The right angle is at B. The angle of elevation at A is θ, the horizontal distance AB is d, and the tower height BC is h.

Select the correct trigonometric ratio

Using SOH-CAH-TOA from angle θ at A: the side opposite θ is the height h (BC), and the side adjacent to θ is the horizontal distance d (AB). Therefore tan(θ) = opposite/adjacent = h/d. Rearranging: h = d · tan(θ). If the hypotenuse (line of sight) is given instead, use sin(θ) = h / hypotenuse, giving h = hypotenuse · sin(θ).

Solve algebraically, then compute

Isolate the unknown variable before substituting numbers. This prevents arithmetic errors and keeps the method visible. After isolating the variable, substitute the given values and evaluate using a calculator or trigonometric tables. Always check that the magnitude of your answer is physically reasonable — a building's calculated height of 2000 metres would signal an error.

Two-position problems

When the angle of elevation is observed from two different positions (say distances d₁ and d₂ from the base), two equations are formed. Subtracting or combining them eliminates the unknown height, allowing the distance to be found first. For example, if angles α and β are observed from points that are x metres apart along the same line: h = x · tan(α)·tan(β) / (tan(α) − tan(β)). This powerful formula appears frequently in competitive examinations.

The Key Equation

h = d \cdot \tan(\theta)

h — Height (metres (or any length unit)): The vertical height of the object above the observer's eye level
d — Horizontal distance (metres (or any length unit)): The horizontal distance measured along the ground from the observer to the base of the object
\theta — Angle of elevation (degrees or radians): The angle between the horizontal line of sight and the line of sight directed toward the top of the object

Worked Example

Problem. A student standing 80 m from the base of a lighthouse observes the top of the lighthouse at an angle of elevation of 35°. The student's eyes are 1.6 m above the ground. Find the total height of the lighthouse above the ground.

H = 80\tan(35^{\circ}) + 1.6 \approx 57.6 \text{ m}

Answer. The lighthouse is approximately 57.6 metres tall.

Figure

Diagram

Origins · 1 minAryabhata and how this was discovered

Aryabhata

476–550 · Indian

Aryabhata was the first mathematician to systematically define and tabulate the sine function (which he called jya) in his landmark work Aryabhatiya (499 CE). His half-chord tables, accurate to four decimal places, laid the foundation for all subsequent trigonometric calculation, including the heights-and-distances techniques still taught today. He applied these methods to calculate the heights of celestial objects and the diameter of the Earth. Aryabhata correctly calculated the value of π to 3.1416 and proposed that the Earth rotates on its own axis — more than a thousand years before Copernicus presented a heliocentric model in Europe.

Trigonometric Ratios in Right Triangles — Heights and distances problems are solved entirely using sine, cosine, and tangent ratios defined in right triangles — mastering SOH-CAH-TOA is the essential prerequisite.
Angles of Elevation and Depression — These two angle types are the primary inputs in every heights and distances problem; understanding how they are measured from the horizontal is fundamental.
The Sine Rule — When the triangle formed in a heights-and-distances scenario is not a right triangle, the Sine Rule (a/sin A = b/sin B = c/sin C) extends the technique to oblique triangles.
The Cosine Rule — The Cosine Rule is needed for two-observation problems where the two observation points and the base of the object do not lie in a straight line, forming a non-right triangle.