Geography & Earth

Distances, maps, coordinates and the planet

The Geography & Earth category brings together practical formulas for measuring the planet, from map scales and coordinates to distances, slopes, density, and time zones. It’s useful for students, exam revision, fieldwork, navigation, and real-world planning, helping you turn geographic data into clear calculations. These formulas matter because they make it easier to understand how the Earth is shaped, mapped, and measured.

Haversine Distance

d = 2r\arcsin\sqrt{\sin^2\frac{\Delta\phi}{2} + \cos\phi_1\cos\phi_2\sin^2\frac{\Delta\lambda}{2}}

Great-circle distance between two lat/long points.

Map Scale

\text{Scale} = \frac{\text{map distance}}{\text{ground distance}}

Ratio of map distance to real-world distance.

Slope / Gradient

\text{gradient} = \frac{\text{rise}}{\text{run}} \times 100\%

Steepness of terrain as a percentage.

Earth Curvature Drop

h \approx \frac{d^2}{2R}

Approximate drop of the horizon over distance d.

Barometric Formula

P = P_0 e^{-\frac{Mgh}{RT}}

Atmospheric pressure as a function of altitude.

Degrees to Distance

1^\circ \text{lat} \approx 111\ \text{km}

Approximate ground distance per degree of latitude.

Richter Magnitude

M = \log_{10}\frac{A}{A_0}

Earthquake magnitude from seismograph amplitude.

Population Density

D = \frac{P}{A}

Number of people per unit area.

Time Zone Difference

\Delta t = \frac{\Delta \text{longitude}}{15^\circ}

Each 15 degrees of longitude equals one hour.

Population Density

D = \frac{P}{A}

People per unit area.

Time Zone Offset

\Delta t = \frac{\Delta\text{longitude}}{15\degree}

Hours difference from longitude difference.

Slope Percent

\text{slope} = \frac{rise}{run}\times 100

Steepness of terrain as a percentage.

Relative Humidity

RH = \frac{e}{e_s}\times 100

Water vapor as a fraction of the saturation amount.

Wind Chill (approx)

WC = 13.12 + 0.62T - 11.37 v^{0.16} + 0.396 T v^{0.16}

Perceived temperature from wind and air temperature.

Dew Point (approx)

T_d \approx T - \frac{100 - RH}{5}

Temperature at which air becomes saturated.

Real Distance from Map

D = d\times S

Actual distance from map distance and scale.

Coriolis Parameter

f = 2\Omega\sin\phi

Strength of the Coriolis effect by latitude.

Dry Adiabatic Lapse Rate

\Gamma \approx 9.8\,\degree C/km

Rate air cools with altitude when rising dry.

Barometric Altitude

h = 44330\left(1 - \left(\frac{P}{P_0}\right)^{0.19}\right)

Altitude estimated from pressure ratio.

Earthquake Energy (Richter)

\log_{10} E = 4.8 + 1.5\,M

Energy released by a quake of Richter magnitude M.

Moment Magnitude

M_w = \tfrac{2}{3}\log_{10} M_0 - 10.7

Modern magnitude from the seismic moment of a quake.

Seismic Wave Speed

v = \sqrt{\frac{E}{\rho}}

Speed of an elastic wave through rock.

Lithostatic Pressure

P = \rho g h

Pressure from the weight of overlying rock.

Geothermal Gradient

\Delta T = G_T \, d

Temperature rise with depth into the Earth.

Tsunami Wave Speed

v = \sqrt{g\,d}

Speed of a shallow-water wave across the ocean.

Radiometric Dating Age

t = \frac{1}{\lambda}\ln\!\left(1 + \frac{D}{P}\right)

Age of a rock from parent and daughter isotopes.

Schumann Resonance

f_1 \approx \frac{c}{2\pi R_E}

Fundamental electromagnetic resonance of the Earth.

Isostatic Root Depth

r = \frac{\rho_c\,h}{\rho_m - \rho_c}

Depth of a mountain's crustal root floating in the mantle.

Darcy's Law

Q = -KA\frac{\Delta h}{L}

Rate of groundwater flow through porous rock.