---
title: "CNC Machinist Math Cheat Sheet: The Formulas That Matter"
description: "A CNC machinist math cheat sheet covers the formulas you actually use: speeds and feeds, basic trig, and bolt circles. Here is what to include and the real limit."
url: https://gcodepractice.com/journal/cnc-machinist-math-cheat-sheet/
canonical: https://gcodepractice.com/journal/cnc-machinist-math-cheat-sheet/
author: "Lawrence Arya"
authorUrl: https://www.linkedin.com/in/vibecoding/
published: 2026-06-08
updated: 2026-06-08
category: "Code reference"
tags: ["machinist math", "cheat sheet", "feeds and speeds", "reference"]
lang: en
---

# CNC Machinist Math Cheat Sheet: The Formulas That Matter

> **TL;DR** A useful CNC machinist math cheat sheet covers the handful of formulas you reach for daily: spindle speed from surface speed, feedrate from chip load, basic right-triangle trig for tapers and angles, and bolt-circle coordinates. Group them by job, keep it to one page, and laminate it. It speeds up the math, but a sheet is a reference, not understanding, so pair it with practice on both the math and the codes.

A CNC machinist does more arithmetic than most people expect, and a good math cheat sheet turns that from a stumbling block into a quick lookup. The trick is knowing which formulas actually earn a place on the sheet. You do not need a textbook on the wall; you need the handful of calculations that come up at the machine every day, laid out so you can grab the right one in seconds.

This guide covers what belongs on a machinist math cheat sheet, the formulas grouped by the job they do, how to format it, and the honest limit of any reference. The math and the codes are the two halves of CNC work, and a cheat sheet helps with the first while leaving the second to memory and practice.

## What belongs on the sheet

A machinist math sheet should cover four areas: cutting parameters, basic trig, hole and circle geometry, and a few handy conversions. That is most of the daily math. Anything rarer you can look up properly when it comes up, which keeps the sheet short enough to actually read. The most-used formulas relate to [speeds and feeds](https://en.wikipedia.org/wiki/Speeds_and_feeds), because every cut needs a spindle speed and a feedrate, and getting them roughly right is the difference between a clean cut and a broken tool.

## Speeds and feeds, the daily math

These are the formulas you reach for most. They convert between the surface speed a material wants and the spindle speed and feedrate your machine needs.

| Formula | What it gives | Notes |
| --- | --- | --- |
| RPM = (SFM x 3.82) / diameter | Spindle speed from surface speed | Diameter in inches; 3.82 is 12 divided by pi |
| Feedrate = RPM x chip load x flutes | Feed in inches per minute | Chip load is per tooth |
| SFM = (RPM x diameter) / 3.82 | Surface speed from RPM | Useful to check a known RPM |
| Chip load = feedrate / (RPM x flutes) | Load per tooth | Confirms you are in a safe range |

The metric versions use the same logic with different constants, where surface speed is in meters per minute and diameter in millimeters. Keep both on the sheet if you run in both systems. These formulas are approximations to get you in range; the material and tooling data refine them, so treat the sheet as a starting point, not a guarantee.

## Basic trig for angles and tapers

The second group is right-triangle trig, which shows up whenever you program an angle, a chamfer, or a taper. You rarely need more than sine, cosine, and tangent.

| Need | Formula |
| --- | --- |
| Opposite side | opposite = hypotenuse x sin(angle) |
| Adjacent side | adjacent = hypotenuse x cos(angle) |
| Angle from two sides | angle = arctangent(opposite / adjacent) |
| Taper radial change | radius change = length x tan(angle from centerline) |

The taper line is the one machinists use constantly on the lathe, and it is worth understanding rather than just copying, because a wrong reference angle cuts the wrong slope. If trig feels intimidating, it is more approachable than it looks, as covered in [whether CNC needs trigonometry](/journal/do-i-need-to-know-trigonometry-to-learn-cnc-programming); the [broader subject](https://en.wikipedia.org/wiki/Trigonometry) is large, but the machinist slice is small.

## Bolt circles and hole patterns

The third group handles holes arranged on a circle, a common feature. For a bolt circle with a given number of holes on a given diameter, each hole sits at an angle, and its X and Y come from trig. The X offset from center is the bolt circle radius times the cosine of the hole angle, and the Y offset is the radius times the sine. Starting angle plus the spacing between holes, which is 360 divided by the number of holes, gives each position. A cheat sheet that lists this pattern saves real time, and it pairs naturally with knowing the codes that position the tool, the same core set worth putting on a [color coded reference card](/journal/color-coded-g-code-reference-card).

## A few conversions worth printing

The last group is small but saves constant fumbling: inch to millimeter and back, decimal equivalents of common fractions, and degrees in a circle. These are quick references rather than formulas, but they belong on the sheet because they come up often and are easy to get slightly wrong under pressure.

## How to format the sheet

A math sheet lives at the machine, so design it to be read fast and survive the shop.

| Choice | Why it matters |
| --- | --- |
| Group by job, not by topic | You find a formula by the task you are doing |
| One page, the common formulas only | Stays scannable, not a textbook |
| Worked example beside each formula | Shows how to plug in the numbers |
| Lamination | Survives coolant, oil, and handling |
| Both inch and metric if you use both | No mental conversion mid-job |

Building your own lets you include the constants and units your shop actually uses, which is the same tailoring that makes a [shop wall poster](/journal/printable-g-code-poster-for-shop-wall) or a [milling visual guide](/journal/visual-guide-to-cnc-milling-poster-pdf) effective. A worked example beside each formula is the single most useful addition, because it removes the doubt about what goes where.

## A worked speeds and feeds example

Numbers make the formulas concrete. Say you are milling aluminum with a half-inch four-flute endmill, and the material runs at roughly 600 surface feet per minute. The spindle speed is 600 times 3.82 divided by 0.5, which is about 4,584 RPM. With a chip load of 0.002 inch per tooth, the feedrate is 4,584 times 0.002 times 4, which is about 36.7 inches per minute. Those two numbers, the S value and the F value, go straight into your program. A reliable reference such as the [CNCCookbook code and machining reference](https://www.cnccookbook.com/g-code-m-code-reference-list-cnc-mills/) lists sensible starting chip loads and surface speeds by material, which is exactly the data that turns the formula into a real number. Always treat the result as a starting point and adjust for the actual cut, the rigidity of the setup, and the sound of the machine.

## Where the math meets the code

The reason a machinist needs both math and code fluency is that the math produces values the code carries. Your calculated spindle speed becomes the S word, your feedrate becomes the F word, and your bolt-circle coordinates become the X and Y positions in the program. So a feedrate you calculated wrong becomes a wrong F in the [G-code](https://en.wikipedia.org/wiki/G-code), and a misread coordinate becomes a hole in the wrong place. The math and the code are not separate worlds; the math sets the numbers and the code delivers them to the machine. This is why a math cheat sheet alone is half the picture: you also have to read the program well enough to confirm those numbers landed where you meant them to.

## The honest limit of a cheat sheet

A math sheet is a genuine time-saver, but it has the same ceiling as any reference: it gives you the formula, not the understanding. If you do not grasp why RPM rises as diameter falls, a wrong number will not look wrong to you, and a cheat sheet cannot catch that. The machinists who use a sheet well are the ones who understand the math underneath it and use the sheet only to skip the memorization of constants.

This matters because the math interacts with the codes. A perfect feedrate calculation still has to go into a correctly written program, and reading that program back to check it is a separate skill. A cheat sheet handles the arithmetic; it does nothing for the code side, which is where a different kind of practice comes in.

## The math mistakes a sheet does not catch

A cheat sheet gives the formula but cannot stop you from misusing it, so a few errors recur. The most common is unit confusion: mixing inch and metric constants, or reading a diameter in millimeters into an inch formula, which throws the result off badly. Another is using a surface speed or chip load meant for a different material, since aluminum and steel run at very different numbers. A third is forgetting that the formulas are starting points, not exact answers, and trusting a calculated feed without listening to how the cut actually sounds. None of these show up as an obviously wrong number on the page, which is why understanding the math matters as much as having the sheet. The machinists who avoid these errors are the ones who know roughly what a sensible answer looks like, so a result that is ten times too high jumps out at them immediately.

## Pair the math with code practice

The math and the codes are the two halves of CNC programming. A cheat sheet covers the math reference well, but the codes you have to read fluently, and that comes from active recall, not a printed sheet. Once your math reference is set, the other half is knowing the G and M codes cold so you can write and verify the program your math feeds into.

The free G-Code Sprint app at GCodePractice.com runs 60-second rounds on the common G and M codes and repeats whatever you miss, building the code-reading fluency that complements your math sheet. It is an educational practice tool for the code side, not a machine controller and not a math solver, so use it alongside the sheet rather than instead of it. With the formulas on the wall and the codes in your head, both halves of the job are covered.

## Frequently asked questions

### What should a CNC machinist math cheat sheet include?

The formulas you use daily, grouped by job: speeds and feeds (spindle speed from surface speed, feedrate from chip load), basic right-triangle trig for angles and tapers, bolt-circle coordinates, and a few inch-to-millimeter conversions. Keep it to one page with a worked example beside each formula. For the code half of the job, the free G-Code Sprint app at GCodePractice.com drills the common G and M codes in 60-second recall rounds.

### How do you calculate RPM for CNC?

Use RPM equals surface speed in SFM times 3.82 divided by the tool diameter in inches. The 3.82 is twelve divided by pi. The metric version uses surface speed in meters per minute and diameter in millimeters with its own constant. The result is a starting point that material and tooling data refine.

### How do you find feedrate on a CNC machine?

Multiply the spindle speed in RPM by the chip load per tooth and by the number of flutes, which gives the feedrate in inches per minute. Chip load is the recommended material removal per tooth, so the formula scales feed to the tool and the cut.

### How do you calculate bolt circle coordinates?

For each hole, the X offset from center is the bolt circle radius times the cosine of the hole angle, and the Y offset is the radius times the sine. The holes are spaced 360 degrees divided by the number of holes, starting from your chosen start angle.

### Is a math cheat sheet enough to learn machinist math?

No. A sheet speeds up the lookup but does not teach the understanding, and a wrong number will not look wrong if you do not grasp the math. Use the sheet to skip memorizing constants, and learn the reasoning so you can catch errors.

### What math do CNC machinists actually use?

Mostly speeds and feeds calculations, basic right-triangle trig for angles and tapers, bolt-circle and hole-pattern geometry, and unit conversions. Advanced math is rare in daily work, so a focused cheat sheet covers most of what comes up.

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Source: https://gcodepractice.com/journal/cnc-machinist-math-cheat-sheet/
Author: Lawrence Arya — https://www.linkedin.com/in/vibecoding/