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Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/physics/measurement/dimension/en.mdx

Output docs content for large language models.

---

import { DimensionLab } from "@repo/design-system/components/contents/physics/measurement/dimension/lab";

export const metadata = {
  title: "Dimensions",
  description:
    "Learn physical dimensions as codes built from base quantities, how to derive dimensions of derived quantities, and how to check formulas without numbers.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/25/2026",
  subject: "Measurement in Scientific Work",
};

## 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.

<BlockMath math="\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.

## Seeing Dimension Powers

The visual below focuses only on the length dimension. Switch from length to area and then volume. Notice how one length factor becomes <InlineMath math="[\mathrm{L}]" />, <InlineMath math="[\mathrm{L}]^2" />, and <InlineMath math="[\mathrm{L}]^3" />.

<DimensionLab
  title={<>Length Dimension Builder</>}
  description={
    <>
      Choose a shape built from length factors to see its formula, unit, and
      dimension.
    </>
  }
  labels={{
    chooseMode: "Choose a dimension shape",
    dimension: "Dimension",
    formula: "Formula",
    modes: {
      length: "Length",
      area: "Area",
      volume: "Volume",
    },
    unit: "SI unit",
  }}
/>

## The Alphabet of Base Quantities

The International System of Units or SI uses <InlineMath math="7" /> base quantities. In dimensional analysis, these base quantities work like an alphabet for building other quantities.

| Base quantity | Example symbol | Dimension |
| :------------ | :------------- | :-------- |
| Length | <InlineMath math="\ell,\ x,\ r" /> | <InlineMath math="[\mathrm{L}]" /> |
| Mass | <InlineMath math="m" /> | <InlineMath math="[\mathrm{M}]" /> |
| Time | <InlineMath math="t" /> | <InlineMath math="[\mathrm{T}]" /> |
| Electric current | <InlineMath math="I" /> | <InlineMath math="[\mathrm{I}]" /> |
| Thermodynamic temperature | <InlineMath math="T" /> | <InlineMath math="[\Theta]" /> |
| Amount of substance | <InlineMath math="n" /> | <InlineMath math="[\mathrm{N}]" /> |
| Luminous intensity | <InlineMath math="I_v" /> | <InlineMath math="[\mathrm{J}]" /> |

Dimensions are written in square brackets so they are not confused with units. For example, <InlineMath math="[\mathrm{L}]" /> is the dimension of length, while <InlineMath math="\text{m}" /> is the unit meter.

## Deriving Dimensions from Formulas

Dimensional analysis works by replacing every quantity in a formula with its dimension, then simplifying the powers.

<BlockMath math="\begin{aligned}
[A] &= [l][w] = [\mathrm{L}][\mathrm{L}] = [\mathrm{L}]^2 \\
[V] &= [l][w][h] = [\mathrm{L}]^3 \\
[v] &= \frac{[\Delta s]}{[\Delta t]} = \frac{[\mathrm{L}]}{[\mathrm{T}]} = [\mathrm{L}][\mathrm{T}]^{-1} \\
[a] &= \frac{[\Delta v]}{[\Delta t]} = \frac{[\mathrm{L}][\mathrm{T}]^{-1}}{[\mathrm{T}]} = [\mathrm{L}][\mathrm{T}]^{-2}
\end{aligned}" />

Force, work, and power can also be read as dimension structures.

<BlockMath math="\begin{aligned}
[F] &= [m][a] = [\mathrm{M}][\mathrm{L}][\mathrm{T}]^{-2} \\
[W] &= [F][\Delta s] = [\mathrm{M}][\mathrm{L}]^2[\mathrm{T}]^{-2} \\
[P] &= \frac{[W]}{[t]} = [\mathrm{M}][\mathrm{L}]^2[\mathrm{T}]^{-3}
\end{aligned}" />

## A Formula That Passes Inspection

Dimensions can help check whether a formula form could be correct. Every term being added must have the same dimension.

Take the displacement formula for uniformly accelerated motion:

<BlockMath math="s = v_0 t + \frac{1}{2} a t^2" />

Check the dimensions:

<BlockMath math="\begin{aligned}
[v_0t] &= [\mathrm{L}][\mathrm{T}]^{-1}[\mathrm{T}] = [\mathrm{L}] \\
[at^2] &= [\mathrm{L}][\mathrm{T}]^{-2}[\mathrm{T}]^2 = [\mathrm{L}]
\end{aligned}" />

Both terms on the right have dimension <InlineMath math="[\mathrm{L}]" />, so the formula passes the dimensional check. This check does not prove the formula is definitely correct, but it can quickly catch impossible formulas.

<BlockMath math="s + t \quad \text{is not valid because} \quad [\mathrm{L}] \ne [\mathrm{T}]" />

## One Object, Many Dimensions

A bolt and a nut look like small objects, but measuring them can involve several different quantities. That is why one object sometimes needs more than one measuring tool.

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

So, two measuring tools can measure quantities with the same dimension, but they serve different contexts. A ruler is enough for a large length that does not need high precision. A vernier caliper or micrometer screw gauge is better when a small diameter must be read more carefully.

OpenStax's concept reference for dimensional analysis can be opened through [this source link](https://openstax.org/books/university-physics-volume-1/pages/1-4-dimensional-analysis).