Formulation
The hypsometric equation is expressed as:

where:
= thickness of the layer [m],
= geometric height [m],
= specific gas constant for dry air,
= mean temperature in kelvins [K],
= gravitational acceleration [m/s2],
= pressure [Pa].
In meteorology,
and
are isobaric surfaces. In altimetry with the International Standard Atmosphere the hypsometric equation is used to compute pressure at a given height in isothermal layers in the upper and lower stratosphere.
Derivation
The hydrostatic equation:

where
is the density [kg/m3], is used to generate the equation for hydrostatic equilibrium, written in differential form:

This is combined with the ideal gas law:

to eliminate
:

This is integrated from
to
:

R and g are constant with z, so they can be brought outside the integral.
If temperature varies linearly with z (as it is assumed to do in the International Standard Atmosphere),
it can also be brought outside the integral when replaced with
, the average temperature between
and
.

Integration gives

simplifying to

Rearranging:

or, eliminating the natural log:
