# ASTM E112 grain size number calculator

> ASTM grain size number G from mean grain area, a grain count or the mean intercept length, and back from G to grain area, intercept and diameter. Free.

URL: https://www.labcanvas.com/tools/astm-grain-size-calculator

The ASTM E112 grain size number is G = 3.321928·log₁₀(N_A) − 2.954, with N_A the number of grains per mm² (1 ÷ the mean grain area in mm²), or G = −6.643856·log₁₀(ℓ̄) − 3.288 from the mean lineal intercept ℓ̄ in mm. Both follow from the definition N = 2^(G − 1) grains per square inch at 100×, so a higher G means smaller grains: a mean intercept of 20 µm is G 8.0. Enter a mean grain area from image analysis, a grain count, an intercept length or G itself.

## Quick answer

- **What it does:** Calculates the ASTM E112 grain size number G from mean grain area, a grain count or the mean intercept length, and converts G back to grain area and diameter.
- **Formula:** G = 3.321928 × log₁₀(N_A) − 2.954 (N_A grains/mm²); G = −6.643856 × log₁₀(ℓ̄) − 3.288 (ℓ̄ in mm); N = 2^(G − 1) per in² at 100×.
- **Example:** Test lines totalling 2 mm cross 100 grains, so ℓ̄ = 0.020 mm and G = −6.643856 × log₁₀(0.020) − 3.288 = 8.0.

Free, in the browser, no account needed. Updated 2026-11-08.

## Features

- Planimetric: G from the mean grain area in nm², µm² or mm², as image analysis reports it
- Planimetric count: grains inside a known area plus half of those cut by its edge
- Intercept (Heyn) method: G from the mean lineal intercept, or from the test-line length and the grains it crosses
- The definition: G from grains per square inch at 100×
- Reverse: G to grains per mm², mean grain area, equivalent circle diameter, √area and mean intercept
- A note when grains are too small for G to be the useful number (nanocrystalline films)

## How to use it

1. **Choose what you measured**: Mean grain area (from Analyze Particles or a grain count in an image analysis program), a count in a known area, a mean intercept, grains per square inch at 100×, or a G to convert back.
2. **Enter it in specimen units**: Type the area or length as measured on the specimen in a scaled image (nm, µm or mm), not on a print.
3. **Read G**: Read G to one decimal and unrounded, with the planimetric or intercept relation it came from.
4. **Read the equivalents**: Read grains per mm², the mean grain area, the equivalent diameter and the mean intercept for the same G.
5. **Report**: Give G with the method (planimetric or intercept), the number of grains or intercepts and fields measured, and for fine grains the mean area and equivalent diameter.

## At a glance

| | |
| --- | --- |
| Formulas | \(G = 3.321928 \cdot \log_{10}(N_\mathrm{A})\) \({} - 2.954\); \(G = -6.643856 \cdot \log_{10}(\bar{\ell})\) \({} - 3.288\) (\(\bar{\ell}\) in mm) |
| Definition | \(N = 2^{G - 1}\) grains per in² at 100×; × 15.50 = grains per mm² at 1× |
| Input | Mean grain area (nm², µm², mm²), a count in a known area, mean intercept or test line and count, N at 100×, or G |
| Output | \(G\), grains per mm², mean area, equivalent diameter, √area, mean intercept, \(N_\mathrm{L}\) |
| Standard | ASTM E112 (current edition E112-25); relations as printed in E112-13 |
| Assumes | Equiaxed grains of one phase on a 2D section, enough grains over several fields |

## How it works

$$N = 2^{G - 1}\ \text{grains per in}^2\ \text{at } 100\times$$

$$N_\mathrm{A} = 15.50 \times N\ \text{grains per mm}^2\ \text{at } 1\times$$

$$G = 3.321928 \cdot \log_{10}(N_\mathrm{A}) - 2.954$$

$$\begin{aligned} G &= -6.643856 \cdot \log_{10}(\bar{\ell}) - 3.288 \\ &= 6.643856 \cdot \log_{10}(N_\mathrm{L}) - 3.288 \end{aligned}$$

**The definition.** ASTM E112, Standard Test Methods for Determining Average Grain Size, defines the grain size number \(G\) through \(N\), the number of grains per square inch at 100× magnification: \(N = 2^{G - 1}\). Multiplying by 15.50 (\(100^2 \div 645.16\) mm² per square inch) gives \(N_\mathrm{A}\), the grains per mm² at 1×, and taking logarithms gives the planimetric relation above, with \(3.321928 = 1/\log_{10}2\). Each step of 1 in \(G\) halves the mean grain area, so a higher \(G\) means finer grains.

**Planimetric method.** Count the grains per unit area of the specimen: in Jeffries’ count, the grains completely inside a test area plus half of those cut by its boundary, divided by the true area. Image analysis gives the mean grain area \(\bar{A}\) of the grains not cut by the frame directly, and for a single-phase structure \(N_\mathrm{A} = 1/\bar{A}\). A mean area of 39,340 nm² is \(3.934\times10^{-8}\ \mathrm{mm^2}\), so \(N_\mathrm{A} = 2.542\times10^{7}\) grains per mm².

**Intercept (Heyn) method.** Lay test lines of known total length \(L\) on the image and count \(N\), the grains they intercept (or \(P\), the grain boundaries they cross). The mean lineal intercept is \(\bar{\ell} = L/N\), in mm at the specimen, and \(N_\mathrm{L} = 1/\bar{\ell}\). \(G = 0\) corresponds to \(\bar{\ell} = 0.32\ \mathrm{mm}\), and halving \(\bar{\ell}\) adds 2 to \(G\) (\(6.643856 = 2/\log_{10}2\)).

**Converting back.** For a given \(G\) the calculator gives \(N_\mathrm{A}\), the mean grain area \(1/N_\mathrm{A}\), its square root (the average diameter of the E112 tables), the diameter of a circle with that area, \(\sqrt{4\bar{A}/\pi}\), and the mean intercept from the intercept relation. The two relations agree only for uniform, equiaxed grains, for which they make \(\bar{\ell} = 0.891 \cdot \sqrt{\bar{A}}\); a real structure measured both ways gives \(G\) values that differ by a fraction of a unit.

**Constants and edition.** The relations are those printed in ASTM E112 (checked against the E112-13 text) and in Metkon’s guide to E112 ([metkon.com](https://www.metkon.com/what-is-astm-e112-a-practical-guide-to-measuring-average-grain-size/)). Some references print the intercept relation as \(G = -6.6457 \cdot \log_{10}\bar{\ell} - 3.298\) (G. F. Vander Voort, [vacaero.com](https://vacaero.com/?p=7854)), which differs by 0.01 \(G\) or less for intercepts from 1 nm to 1 mm. The current edition is ASTM E112-25 ([store.astm.org](https://store.astm.org/e0112-25.html)); consult the standard for the full procedure: specimen preparation, the comparison, planimetric and intercept procedures, how many fields to measure and the confidence interval. ASTM’s summary of E112-13 (reapproved 2021) states that a precision of ±0.25 grain size units can be reached with a reasonable amount of effort.

**Assumptions and limits.** E112 measures planar grain size, on a two-dimensional section, and applies chiefly to single-phase structures; the relations here are for equiaxed grains, and the standard treats non-equiaxed grains (elongated, partly recrystallised or cold-worked) separately, measuring in more than one direction. Duplex structures, with two grain size populations, are covered by ASTM E1181. Count enough grains, over several fields, for the mean to be representative. Nanocrystalline films give very high \(G\): a mean grain area of 39,340 nm² (equivalent diameter 224 nm) is \(G\) 21.6, where the mean grain area and equivalent diameter, with the number of grains measured, tell a reader more. A top-view image of a film is also not the random section the method assumes.

## Worked examples: a perovskite film, a steel and a G table

**1. A perovskite film, from the mean grain area.** The built-in thin-film sample in [LabCanvas Image Analysis](https://www.labcanvas.com/analysis.html) (a synthetic image with a scale in nm) gives 118 grains not cut by the frame, with a mean grain area of 39,340 nm² and an equivalent diameter of 224 nm.

- \(\bar{A} = 39{,}340\ \mathrm{nm^2} = 3.934\times10^{-8}\ \mathrm{mm^2}\), so \(N_\mathrm{A} = 1 \div \bar{A} = 2.542\times10^{7}\) grains per mm².
- \(G = 3.321928 \times \log_{10}(2.542\times10^{7})\) \({}- 2.954\) \(= 3.321928 \times 7.4052 - 2.954\) \(= \mathbf{21.6}\).
- Back from \(G\): \(\sqrt{\bar{A}} = 198\ \mathrm{nm}\) and the mean intercept for the same \(G\) is 177 nm.

\(G = 21.6\) is a correct number but not a useful one: it lies far beyond the table below, which reaches a mean intercept of 2.5 µm at \(G\) 14. Report the mean grain area (39,340 nm²) and equivalent diameter (224 nm) with n = 118, and \(G\) only if a reader asks for it.

**2. A steel, from the mean intercept.** Test lines with a total length of 2 mm on the specimen cross 100 grains, so \(\bar{\ell} = 2\ \mathrm{mm} \div 100 = 0.020\ \mathrm{mm}\) (20 µm).

- \(G = -6.643856 \times \log_{10}(0.020) - 3.288\) \(= -6.643856 \times (-1.69897) - 3.288\) \(= \mathbf{8.0}\).
- The same \(G\) from the definition: \(N = 2^{8 - 1} = 128\) grains per square inch at 100×, \(N_\mathrm{A} = 15.50 \times 128 = 1984\) grains per mm², a mean grain area of 504 µm² and an equivalent diameter of 25.3 µm.

**3. A planimetric count.** 95 grains lie completely inside a 0.5 mm² field and 46 are cut by its edge: \(N_\mathrm{A} = (95 + 46/2) \div 0.5 = 236\) grains per mm², \(G = 3.321928 \times \log_{10}(236) - 2.954\) \(= \mathbf{4.9}\).

| G | Grains per in² at 100× | Grains per mm² | Mean area (µm²) | Equivalent diameter (µm) | Mean intercept (µm) |
| --- | --- | --- | --- | --- | --- |
| 0 | 0.5 | 7.75 | 129,000 | 405 | 320 |
| 1 | 1 | 15.5 | 64,500 | 287 | 226 |
| 2 | 2 | 31.0 | 32,300 | 203 | 160 |
| 3 | 4 | 62.0 | 16,100 | 143 | 113 |
| 4 | 8 | 124 | 8,070 | 101 | 80.0 |
| 5 | 16 | 248 | 4,030 | 71.7 | 56.6 |
| 6 | 32 | 496 | 2,020 | 50.7 | 40.0 |
| 7 | 64 | 992 | 1,010 | 35.8 | 28.3 |
| 8 | 128 | 1,984 | 504 | 25.3 | 20.0 |
| 9 | 256 | 3,967 | 252 | 17.9 | 14.1 |
| 10 | 512 | 7,935 | 126 | 12.7 | 10.0 |
| 11 | 1,024 | 15,870 | 63.0 | 8.96 | 7.07 |
| 12 | 2,048 | 31,740 | 31.5 | 6.33 | 5.00 |
| 14 | 8,192 | 127,000 | 7.88 | 3.17 | 2.50 |

## Questions

### How do you calculate the ASTM grain size number?

From the grains per mm² at 1×, \(N_\mathrm{A}\): \(G = 3.321928 \times \log_{10}(N_\mathrm{A}) - 2.954\). From the mean lineal intercept \(\bar{\ell}\) in mm: \(G = -6.643856 \times \log_{10}(\bar{\ell}) - 3.288\). Both come from the definition \(N = 2^{G - 1}\) grains per square inch at 100×.

### How do I get G from a mean grain area measured in ImageJ or another image analysis program?

Convert the mean grain area to mm², take \(N_\mathrm{A} = 1 \div \text{area}\), then \(G = 3.321928 \times \log_{10}(N_\mathrm{A}) - 2.954\). A mean area of 504 µm² (\(5.04\times10^{-4}\ \mathrm{mm^2}\)) gives \(N_\mathrm{A} = 1984\) per mm² and \(G = 8.0\). Leave out grains cut by the image frame.

### What is the difference between the planimetric and intercept methods?

The planimetric (Jeffries) method counts grains per unit area; the intercept (Heyn) method counts how many grains test lines of known length cross and gives the mean intercept length. For uniform equiaxed grains they give the same G; for real structures they differ by a fraction of a unit.

### Does a higher grain size number mean larger or smaller grains?

Smaller. Each step of 1 in G doubles the number of grains per unit area and halves the mean grain area; each step of 2 halves the mean intercept. G 8 has a mean intercept of 20 µm, G 10 of 10 µm.

### How do I convert an ASTM grain size number to a grain diameter in µm?

Say which diameter: for G 8 the mean intercept is 20.0 µm, the diameter of a circle with the mean grain area is 25.3 µm, and the square root of the mean area (the E112 tables’ average diameter) is 22.5 µm. The calculator gives all three.

### Can I use ASTM E112 for thin films or nanoparticles?

The arithmetic works, but G comes out very high: grains of a perovskite film with a mean area of 39,340 nm² give G 21.6, a number that tells a reader little. For nanocrystalline films report the mean grain area and equivalent diameter with the number of grains, and remember that a top-view image is not the random section E112 assumes.

### Which edition of ASTM E112 is current?

ASTM E112-25 is the current edition at the time of writing. The relations used here are those printed in E112 (checked against E112-13); consult the standard for the full procedure, the number of fields and the statistics.