Calculate the Width of a Visible Object on the Ground

Calculator for the width or breadth of something like a river or so, which is flat on the ground and is watched from a certain height. This calculation is correct, when the distance to the object and the object itself have a size so that the curvature of the earth is negligible. For rivers with a width of up to a few dozend meters or smaller things, this is the case. The difficulty is measuring the angles, if you don't have a sextant or similar equipment.

The eye height h is the height of the eyes above a flat ground. The horizontal distance l is the distance of the eyes projected vertically to the ground to the closer, front edge of the object. Instead of meters (m), you can also use feet or every other unit of length, as long as it is the same unit for all lengths.

Formulas:
γ = α − β
d = √ h² + l²
b = d * sin(γ) / sin(β)

Calculating the visible width of an object on the ground is based on geometric principles and demonstrates how perspective and distance influence perception. Especially when viewed from an elevated position, such as a tower or hill, the width of an object like a river appears greater because the viewing angle increases. Historically, navigators and surveyors used similar methods to estimate distances long before precise instruments were available. The underlying mathematics is universally applicable, for example in photography and architecture. Such calculations are based on triangles and are therefore called trigonometry. Its best-known methods are sine and cosine. The sine function is used in the formula above to calculate a different length from a length and an angle.

Eye height h:		m
Horizontal distance l:		m
Angle to the front edge α:		°
Angle to the back edge β:		°

Breadth angle γ:		°
Distance eye - object d:		m
Objekt width b:		m