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Angles, Distances, and Trigonometry Example

Easy DifficultyFE Surveying

Reduce a slope distance to horizontal distance and vertical difference using the measured vertical angle.

In surveying, distances are often measured along the slope . For mapping and horizontal positioning, the horizontal distance H is needed. If the vertical angle θ (from horizontal) and slope distance S are known: H = S cos θ and the vertical difference is V = S sin θ . H = S cos (θ) V = S sin (θ) Notation: S = slope distance, H = horizontal distance, V = vertical difference, θ = vertical angle from horizontal.

A total station measures a slope distance of 150.00 ft to a target. The vertical angle from the instrument to the target is 5°30' above horizontal. Find: Horizontal distance between instrument and target. Vertical difference (elevation change) between instrument and target.

S = 150.00 ft (slope distance) θ = 5°3 0^{'} (vertical angle, above horizontal)

H = S cos (θ) H = 150 cos (5°3 0^{'}) = 150 \times 0.9952 \approx 149.3 ft

V = S sin (θ) V = 150 sin (5°3 0^{'}) = 150 \times 0.0958 \approx 14.4 ft

(1) Horizontal distance: H \approx 149.3 ft . (2) Vertical difference: V \approx 14.4 ft (target is higher).

H = S cos θ, V = S sin θ (slope to horizontal)