---
title: "How to Calculate I and J for a G02 Arc (Worked Examples)"
description: "I and J are the distance from the arc's start point to its center: I = center X minus start X, J = center Y minus start Y. Two worked examples and a self-check."
url: https://gcodepractice.com/journal/how-to-calculate-i-and-j-for-g02-arc/
canonical: https://gcodepractice.com/journal/how-to-calculate-i-and-j-for-g02-arc/
author: "Lawrence Arya"
authorUrl: https://www.linkedin.com/in/vibecoding/
published: 2026-06-04
updated: 2026-06-04
category: "Code reference"
tags: ["g02", "arc", "i-j-k", "calculation", "beginner"]
lang: en
---

# How to Calculate I and J for a G02 Arc (Worked Examples)

> **TL;DR** I and J locate the arc center relative to the start point. Calculate them by subtraction: I equals the center's X minus the start point's X, and J equals the center's Y minus the start point's Y, keeping the signs. Check your work with the radius: the square root of I squared plus J squared must equal the arc radius, and the center must sit at that same radius from the endpoint too.

Working out `I` and `J` by hand intimidates beginners, but the entire calculation is one rule: subtract the start point from the center, coordinate by coordinate. Every wrong arc traced back to this calculation comes from subtracting the wrong pair of points or dropping a sign.

## What are I and J measuring?

`I` and `J` describe the straight-line offset from the arc's **start point** to the arc's **center**: `I` along X, `J` along Y. They are a small vector on the ordinary [Cartesian grid](https://en.wikipedia.org/wiki/Cartesian_coordinate_system), and on standard controls they are incremental, measured from the start point, as the [LinuxCNC arc format](https://linuxcnc.org/docs/html/gcode/g-code.html) specifies. (If you are still deciding whether to use this format or the simpler radius word at all, the [R versus I/J overview](/journal/g-code-r-vs-i-and-j-for-beginners/) covers that choice first.) The start point is simply wherever the previous move left the tool.

So the rule is:

```
I = center X - start X
J = center Y - start Y
```

Keep the signs exactly as the subtraction produces them. A center left of the start point gives a negative `I`; a center below gives a negative `J`.

## A worked example, step by step

The tool sits at `X20. Y10.` and the drawing shows a 5 mm radius arc around a center at `X15. Y10.`, ending at `X10. Y10.`:

| Step | Calculation | Result |
| --- | --- | --- |
| Start point | from the previous move | X20, Y10 |
| Center | from the drawing | X15, Y10 |
| `I` | 15 - 20 | -5.0 |
| `J` | 10 - 10 | 0 |
| Radius check | sqrt((-5)^2 + 0^2) | 5.0 ✓ |

The block comes out as:

```
G02 X10. Y10. I-5. J0 F150
```

The negative `I` is doing real work: it says the center lies five units in the negative X direction from where the arc begins.

## A second example with both offsets

Start point `X0 Y0`, center at `X3. Y4.`, full circle back to the start. The subtraction gives `I3. J4.`, and the radius check gives the hypotenuse: the square root of 9 plus 16 is 5, so this is a 5 mm radius circle:

```
G02 X0 Y0 I3. J4. F150   (full circle, center up and to the right)
```

Full circles are the case where this calculation is unavoidable, because the radius word cannot define them, the limitation explained in [I, J, K vs R in G02](/journal/difference-between-i-j-k-and-r-in-g02/). The same start-to-center subtraction, generalized to many points, is how a [bolt hole circle's coordinates](/journal/how-to-find-x-and-y-coordinates-for-bolt-hole-circle-g-code/) get computed with sine and cosine.

## How do you verify the numbers before running?

The radius gives you a two-step self-check that catches nearly every slip. First, `I` and `J` span start-to-center, so the square root of `I` squared plus `J` squared must equal the arc radius. Second, the center must sit at that same radius from the **endpoint**, because both ends of an arc lie on the circle. Run both checks; if they disagree, one of your three inputs (start, end, center) is wrong. Controls enforce this geometry, and an inconsistent arc either alarms or cuts a shape you did not intend, one of the failure paths covered in [why G02 cuts a straight line](/journal/why-is-g02-cutting-a-straight-line/).

## Where do the calculations go wrong?

Three mistakes account for almost all bad `I J` values:

| Mistake | What happens | Guard |
| --- | --- | --- |
| Using the center's absolute coordinates as `I J` | Center lands far from the arc | Always subtract the start point |
| Measuring from the endpoint | Mirrored or shifted center | The start point is the reference |
| Dropped minus sign | Arc bulges the wrong way | Keep signs from the subtraction |

The direction of travel around that center is the separate clockwise question settled by [G02 vs G03](/journal/g02-vs-g03/), and the standard [code references](https://www.cnccookbook.com/g-code-m-code-cnc-list-cheat-sheet/) list both forms of the arc words for quick lookup.

## Bottom line

`I` and `J` are the center minus the start point, sign included: one subtraction for X, one for Y. Verify with the radius twice, start-to-center and center-to-end, and the arc is proven before the machine ever sees it. Making that subtraction rule reflex is a recall job, and a routine on the [G-code practice hub](/g-code-practice/) handles it alongside the codes themselves.

## Sources

- [LinuxCNC G-code reference (G2/G3 incremental arc centers)](https://linuxcnc.org/docs/html/gcode/g-code.html)
- [Wikipedia: Cartesian coordinate system](https://en.wikipedia.org/wiki/Cartesian_coordinate_system)
- [CNCCookbook: G-code and M-code cheat sheet](https://www.cnccookbook.com/g-code-m-code-cnc-list-cheat-sheet/)

## Frequently asked questions

### How do you calculate I and J for a G02 arc?
Subtract the start point from the center: `I` equals center X minus start X, `J` equals center Y minus start Y, signs kept. An arc starting at `X20 Y10` around a center at `X15 Y10` gets `I-5. J0`.

### Are I and J measured from the start point or the end point?
From the start point on standard controls. Computing from the endpoint is the most common mistake and misplaces the center.

### How do you check that I and J are correct?
The square root of `I` squared plus `J` squared must equal the radius, and the center must sit at the same radius from the endpoint. If the two distances disagree, one of the inputs is wrong.

### What is the best way to learn arc calculations like I and J?
Drill the rule with active recall and practice on paper arcs. A free app like G-Code Sprint quizzes `G02`, `G03`, and the arc words and repeats whichever ones you miss.

*G-Code Sprint is a study and practice tool only. Always follow your instructor, employer, machine manual, and shop safety procedures.*

---

Source: https://gcodepractice.com/journal/how-to-calculate-i-and-j-for-g02-arc/
Author: Lawrence Arya — https://www.linkedin.com/in/vibecoding/