# How Does a Barometer Work?

## Answer #1

Level 3 (with higher mathematics)

Answered by Alexander Fufaev

A simple barometer works by using the **barometer formula**, which relates the height \(h\) above the ground to the air pressure \( \mathit{\Pi} \) prevailing at that height:

**Barometric formula**

Formula anchor $$ \begin{align} \mathit{\Pi}(h) ~=~ \mathit{\Pi}_0 \, \mathrm{e}^{-h/H} \end{align} $$

Here \( H ~=~ 8005 \, \mathrm{m} \) and the mean air pressure at sea level is \(\mathit{\Pi}_0 = 101 \, \mathrm{kPa} = 1 \, \mathrm{bar} \). The formula states that the air pressure decreases *exponentially* with increasing height \( h \).

So an barometer measures the air pressure \( \mathit{\Pi} \) and calculates from it the height at which this air pressure prevails:

**Formula: Air pressure**

Formula anchor $$ \begin{align} h ~=~ 8005 \, \mathrm{m} \,\cdot\, \ln\left(\frac{ \mathit{\Pi} }{ 101 000 \, \mathrm{Pa} }\right) \end{align} $$