> ## Documentation Index
> Fetch the complete documentation index at: https://docs.truemath.ai/llms.txt
> Use this file to discover all available pages before exploring further.

# Units

> The catalog of units TrueMath supports across length, area, volume, mass, time, angle, information, and currency, plus compound units and conversion behavior.

TrueMath attaches units to values and converts between compatible units automatically. This page catalogs the supported units by dimension and explains how compound units and conversions behave. For how units are written inside activity equations, see [Writing equations](/authoring/writing-equations); for how unit numbers behave as a data type, see [Unit numbers](/math/types/unit-numbers).

## Simple units

| Dimension   | Units                                                                                                                |
| ----------- | -------------------------------------------------------------------------------------------------------------------- |
| Length      | `m`, `cm`, `mm`, `km`, `in` (`"`), `ft` (`'`), `yd`, `mi`, `nmi` (`NM`)                                              |
| Area        | `m^2`, `cm^2`, `mm^2`, `km^2`, `in^2`, `ft^2`, `yd^2`, `mi^2`, `ha`, `ac`                                            |
| Volume      | `m^3`, `cm^3`, `mm^3`, `km^3`, `in^3`, `ft^3`, `yd^3`, `L`, `ml`, `tsp`, `tbsp`, `floz`, `c`, `pt`, `qt`, `gal`      |
| Mass        | `mg`, `g`, `kg`, `t`, `oz`, `lb`, `tn`                                                                               |
| Time        | `ms`, `s`, `min`, `hr`, `d`, `wk`, `mo`, `yr`                                                                        |
| Angle       | `rad`, `deg`                                                                                                         |
| Information | `bit`, `kbit`, `Mbit`, `Gbit`, `Tbit`, `B`, `kB`, `MB`, `GB`, `TB`, `PB` (and binary forms `Kibit…Tibit`, `KiB…PiB`) |
| Currency    | `USD`                                                                                                                |
| Count       | `ct`                                                                                                                 |

Area and volume units also accept the typographic forms `m²` and `m³`. Time units can also be squared (`s^2`, `hr^2`, …); squared time is the time² denominator that appears in acceleration units such as `m/s^2`.

Most units also accept their full names, singular or plural, including common spelling variants — `ft` can be written `foot` or `feet`, and `m` as `meter`, `meters`, `metre`, or `metres`.

Unit suffixes are case-sensitive: `USD` is not `usd`, and `Gbit` is not `gbit` or `GBIT`. Use the exact casing shown above.

## Compound units

A compound unit combines a unit from one dimension with a unit from another. **Any** simple unit from each constituent dimension can be paired, so velocity covers `m/s`, `km/hr`, `ft/min`, and every other length-over-time combination, not only the named shorthands. TrueMath also recognizes a few shorthand suffixes for common pairings.

| Kind           | Form                                                        | Named shorthands                            |
| -------------- | ----------------------------------------------------------- | ------------------------------------------- |
| Velocity       | length / time                                               | `kph` (km/hr), `mph` (mi/hr), `kt` (nmi/hr) |
| Acceleration   | length / time²                                              | —                                           |
| Flow rate      | volume / time                                               | `GPM` (gal/min)                             |
| Area rate      | area / time                                                 | —                                           |
| Bandwidth      | information / time                                          | —                                           |
| Cost           | currency / length, area, volume, time, mass, or information | —                                           |
| Per-unit count | count / length, area, volume, time, mass, or currency       | —                                           |

Cost and per-unit-count compounds are written directly from their constituent units, for example `USD/hr`, `USD/ft^2`, or `ct/m^2`.

## Conversion behavior

* **Compatible units convert automatically.** `1m + 10cm` produces a length; you never convert by hand.
* **The result takes the first unit in the expression.** `5in + 10ft` is `125in`, and `5hr + 6min` is `5.1hr` — the leading unit sets the result and the rest convert into it.
* **Multiplication and division derive dimensions.** A length over a time is a velocity; a currency over an area is a cost-per-area.
* **Incompatible units are rejected.** `5m + 3kg` is an error rather than a wrong number. See [Combining types](/authoring/writing-equations#combining-types).

### Casting to a specific unit

To choose the unit a result is reported in, append a target unit to a parenthesized expression:

```
(1m + 10cm) mm
```

This returns the sum expressed in millimetres. Casting changes only the reported unit, not the underlying value, and the target must be compatible with the result's dimension.

<Note>
  Angles are interpreted by the unit you attach: `sin(45deg)` and `sin((pi/4)rad)` are equivalent. See [Trigonometry](/math/functions/trigonometry).
</Note>