Surveying & Geomatics

Measuring the land: slope and grade, coordinate areas, stadia, traverse latitudes and map scale

Surveying & Geomatics formulas help you measure and map the land accurately, from slope and grade percentage to coordinate areas, traverse calculations, stadia distance, and map scale. They’re useful for students, exam prep, and professionals working in surveying, construction, civil engineering, and GIS, where precise distances, elevations, closure checks, and curvature corrections are essential for reliable real-world results.

Slope / Grade Percentage

G = \frac{rise}{run} \times 100

Steepness of ground or a road as a percentage.

Area by Coordinates (Shoelace)

NEWIntermediate

A = \frac{1}{2}\left| \sum (x_i y_{i+1} - x_{i+1} y_i) \right|

Area of a polygon from the coordinates of its corners.

Horizontal Distance

H = L \cos\theta

Level distance from a slope distance and its inclination angle.

Vertical Distance

V = L \sin\theta

Elevation change from a slope distance and angle.

Traverse Latitude

NEWIntermediate

Lat = L \cos(\alpha)

North-south component of a traverse line from its length and bearing.

Traverse Departure

NEWIntermediate

Dep = L \sin(\alpha)

East-west component of a traverse line.

Traverse Closure Error

E = \sqrt{(\sum Lat)^2 + (\sum Dep)^2}

Straight-line gap between the start and computed end of a closed traverse.

Relative Precision

NEWIntermediate

RP = \frac{E}{P}

Closure error expressed as a fraction of the traverse perimeter.

Stadia Distance

D = 100 \, s \cos^2\theta

Horizontal distance from stadia hair readings on a level rod.

Map Scale (RF)

RF = \frac{d_{map}}{d_{ground}}

Representative fraction linking map distance to real-world distance.

Earth Curvature Correction

C = 0.0675 \, K^2

Drop due to Earth's curvature over a sight distance (metres, K in km).

Radius from Degree of Curve

R = \frac{5729.58}{D}

Radius of a circular road or rail curve from its degree of curve.

Area by Simpson's Rule

A = \frac{h}{3}(y_0 + 4\Sigma_{odd} + 2\Sigma_{even} + y_n)

Area under an irregular boundary using parabolic strips.

Area by Trapezoidal Rule

NEWIntermediate

A = h\left(\frac{y_0 + y_n}{2} + \Sigma y_i\right)

Area under a boundary by summing trapezoidal strips.

Bearing from Coordinates

NEWIntermediate

\alpha = \arctan\frac{\Delta E}{\Delta N}

Direction of a line from the change in easting and northing.

Slope Between Contours

S = \frac{CI}{d} \times 100

Ground slope from the contour interval and horizontal spacing on a map.