Calculation example - Calculate the height of an object from three angles of elevation

Calculate the height of an object by measuring three angles of elevation only, from stations A, B and C on a straight level line to an inaccessible up-station D.

Known values:

Measured distances AB = x and BC = y.

Vertical angles α, β, θ measured at A, B and C respectively.

Solution: Height of object h.

From vertical triangles ADE, BDE and CDE:

Solving triangles ADB and ADC by the cosine rule:

In triangle ADB,

Same for triangle ADC,

Therefore:

And: