Formula Sheet

Heights & Distances Formulas

All key formulas grouped by subtopic. Each one has a quick reminder and common mistakes to watch for.

10 formulas · 6 subtopics

Basic Height from Angle of Elevation

\tan(\alpha) = \frac{h}{d} \implies h = d \cdot \tan(\alpha)

💡 Here h is the height of the object above the observer's eye level, d is the horizontal distance, and alpha is the angle of elevation.

Angle of Elevation Definition

\text{Angle of Elevation} = \text{angle between horizontal and line of sight (looking UP)}

💡 The angle of elevation is always measured from the horizontal line at the observer's eye level upward to the object. It is never negative.

Angle of Depression Definition

\text{Angle of Depression} = \text{angle between horizontal and line of sight (looking DOWN)}

💡 The angle of depression from point A to point B equals the angle of elevation from B to A (alternate interior angles). This duality is tested frequently.

⚠ Students confuse elevation with depression. Elevation is looking UP, depression is looking DOWN from a higher point.

Height from Two Observation Points

h = \frac{d \cdot \tan(\alpha) \cdot \tan(\beta)}{\tan(\beta) - \tan(\alpha)}

💡 Two points at distance d apart observe the top of a tower at angles alpha and beta (beta > alpha). Both observers are on the same side and at the same level as the base.

⚠ This formula assumes both observers are on the same side of the tower. If on opposite sides, use tan(beta) + tan(alpha) in the denominator.

Distance Between Two Points Using Two Angles

d = h\left(\cot(\alpha) - \cot(\beta)\right)

💡 If a tower of height h is observed from two points on the same side, the distance between them is h(cot(alpha) - cot(beta)) where alpha < beta.

Shadow Length Formula

\text{Shadow length} = h \cdot \cot(\theta) = \frac{h}{\tan(\theta)}

💡 Here theta is the angle of elevation of the sun. As the sun rises (theta increases), the shadow gets shorter. At theta = 45 degrees, shadow length equals height.

Sine Rule in Heights and Distances

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

💡 When the triangle formed is not right-angled (e.g., objects on hills), use the sine rule to find unknown sides or angles.

Height from Moving Observer

h = \frac{d \cdot \tan(\alpha) \cdot \tan(\beta)}{\tan(\beta) - \tan(\alpha)}

💡 Same formula as two-point observation. When a person walks d metres toward a tower and the angle changes from alpha to beta, this gives the height.

Combined Elevation and Depression

\text{Total height} = d \cdot \tan(\alpha) + d \cdot \tan(\beta)

💡 When standing between two objects (or at the top of a building looking up at one and down at another), the total height combines both components.

Height of Object on a Hill

#10

h_{\text{object}} = \frac{d \cdot \sin(\alpha - \beta)}{\cos(\beta)}

💡 When an object of height h stands on a hill inclined at angle beta, and the object subtends angle alpha at a point at the base, use this to find h.

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Mathematical Reasoning