Dimensions

Dimension Is Not Just Object Size

In everyday language, dimension often means spatial size such as length, width, and height. In physics, dimension has a more specific meaning: it is a code that shows which base quantities build a physical quantity.

For example, bolt length, nut diameter, marble radius, and travel distance can use different units, but they are still the same kind of quantity.

\text{length, diameter, radius, distance} \Rightarrow [\mathrm{L}]

A dimension does not tell us the measured value. It tells us the physical kind behind the measured value.

Measurement

Dimensions

Dimension Is Not Just Object Size

For example, bolt length, nut diameter, marble radius, and travel distance can use different units, but they are still the same kind of quantity.

\text{length, diameter, radius, distance} \Rightarrow [\mathrm{L}]

A dimension does not tell us the measured value. It tells us the physical kind behind the measured value.

Base quantity	Example symbol	Dimension
Length	$\ell,\ x,\ r$	$[\mathrm{L}]$
Mass	$m$	$[\mathrm{M}]$
Time	$t$	$[\mathrm{T}]$
Electric current	$I$	$[\mathrm{I}]$
Thermodynamic temperature	$T$	$[\Theta]$
Amount of substance	$n$	$[\mathrm{N}]$
Luminous intensity	$I_v$	$[\mathrm{J}]$

What is checked	Suitable tool	Dimension
Bolt length	ruler or vernier caliper	$[\mathrm{L}]$
Outer diameter	vernier caliper or micrometer screw gauge	$[\mathrm{L}]$
Cross-sectional area	calculated from diameter	$[\mathrm{L}]^2$
Material volume	calculated from spatial size	$[\mathrm{L}]^3$
Mass	balance	$[\mathrm{M}]$
Density	mass divided by volume	$[\mathrm{M}][\mathrm{L}]^{-3}$

Dimensions

Dimension Is Not Just Object Size

Dimensions

Dimension Is Not Just Object Size

The Alphabet of Base Quantities

Deriving Dimensions from Formulas

A Formula That Passes Inspection

One Object, Many Dimensions