What is the true position formula?

True position = 2 × √(Δx² + Δy²), where Δx and Δy are the measured deviations from the basic location in the datum reference frame. The factor of two is essential: position defines a diametral tolerance zone, so a feature 0.25 mm from true position consumes a Ø0.50 mm zone. Reporting the radial distance instead of the diameter is the most common mistake in position reporting.

How do you calculate bonus tolerance at MMC?

Bonus is the departure of the feature from its maximum material condition. For a hole, MMC is the smallest allowed size, so bonus = actual size − MMC size; for a pin, MMC is the largest allowed size, so bonus = MMC size − actual size. Add that to the stated tolerance to get the allowed zone. Bonus is never negative, and a feature outside its own size limits earns none — it is a size reject before position is assessed.

Why is a GD&T position tolerance round instead of a square ± box?

Because a round zone gives about 57% more usable area for the same worst-case deviation. The largest square that fits inside a Ø t circle has a per-axis tolerance of ±t/(2√2), and the circle's area exceeds the square's by a factor of π/2. Features that land in the corner region pass position while failing an equivalent coordinate callout — and they still assemble, which is what the tolerance is protecting.

What is the equivalent ± coordinate tolerance for a Ø0.5 position tolerance?

±0.1768 per axis. That is the largest square zone that fits entirely inside the Ø0.5 circle, so a part meeting it always meets the position callout — but not the other way round, since the circle also accepts everything in the four corner regions the square excludes.

Does this true position calculator handle MMC and LMC?

Yes. Select Ⓜ or Ⓛ, say whether the feature is a hole or a pin, and enter the actual size with both size limits. The calculator works out which limit is the stated material condition, computes the departure as bonus tolerance, adds it to the stated tolerance, and gives the pass or fail against the enlarged zone. Datum shift at MMB is a separate allowance and is not included.

Calculator · GD&T

True Position Calculator — GD&T Position With MMC Bonus

https://shopmath.org/calculators/true-position · data verified August 1, 2026

A feature measured 0.15 mm off in X and 0.20 mm off in Y is at true position Ø0.5000 mm — twice the 0.2500 mm radial deviation, because a position tolerance is a diametral zone. Against a Ø0.5 mm callout that passes, using 100% of the zone. Add the Ⓜ modifier and the feature’s own size can enlarge the zone further.

Standard: ASME Y14.5 · ISO 1101
Modifiers: RFS · MMC · LMC
Units: mm or in
Axes: X, Y
Verified: August 1, 2026

True Position Calculator — GD&T Position From X and Y Deviation

Unit-agnostic: put millimetres in and get millimetres out, or inches and inches. Nothing is converted, so there is no way to mix units and get a plausible wrong answer.

True position calculator

Material condition

Unit

Deviation in Xmm

Measured minus basic. Sign does not affect the result.

Deviation in Ymm

From the same datum reference frame as X.

Position toleranceØ mm

The diametral value in the feature control frame.

True position · regardless of feature size · Ø0.5 mm zone

Pass

Ø0.5000

mm — the zone this feature consumes

Deviation from true: 0.2500mm
Allowed Ø: 0.5000mm
Zone used: 100%
Direction: 53.13°

The feature uses 100% of the allowed zone, leaving Ø0.0000 mm of margin.

Drawn to scale. The solid circle is the stated Ø0.5000 mm tolerance zone. The dashed square is the ±0.1768 mm coordinate tolerance that would fit inside it — the circle has about 57% more area, which is the whole reason position is stated as a diameter.

The formula, with your numbers in it

ASME Y14.5 · ISO 1101

```
true position  =  2 × √( Δx² + Δy² )
```
```
true position  =  2 × √( 0.15² + 0.2² )  =  2 × 0.2500
```
= Ø0.5000 mm
The factor of two is the whole concept: position is a DIAMETRAL zone centred on the theoretically exact location, so a feature 0.2500 mm off centre consumes a zone twice that across. Reporting the radius is the most common way a good part gets scrapped.

allowed Ø  =  stated tolerance + bonus          pass when true position ≤ allowed Ø

allowed Ø  =  0.5 + 0.0000  =  0.5000
0.5000 ≤ 0.5000

= PASS

A Worked True Position Example

The calculator's default state, written out — the answer is on the page whether or not anything runs.

A hole is measured 0.15 mm off in X and 0.20 mm off in Y from its basic location. The radial distance from true position is √(0.15² + 0.20²) = 0.2500 mm, so the true position value is twice that: Ø0.5000 mm.

Against a Ø0.5 mm position tolerance the feature uses 100% of the zone and passes, with Ø0.0000 mm of margin left. Against a tighter Ø0.4 mm callout the same measurement fails — it exceeds the zone by Ø0.1000 mm.

Now add the modifier. If the callout is Ø0.5 Ⓜ, the hole is specified Ø10.0–10.2, and it was produced at Ø10.15, then it has departed 0.1500 mm from its maximum material condition of Ø10.0000. That departure becomes bonus tolerance, the allowed zone grows to Ø0.6500 mm, and the feature now uses only 76.9% of it.

One rule the arithmetic enforces on its own: if that hole had been produced at Ø10.5, outside its own size limits, it would earn no bonus at all. It is a size reject before position is considered, and letting bonus accrue past the limit would let a size failure hide a position failure.

Why GD&T True Position Is Stated as a Diameter

The single most common error in position reporting is handing back the radius. Here is what the factor of two buys.

A ± coordinate tolerance defines a square zone; a position tolerance defines a round one centred on the same point. The largest square that fits inside a Ø t circle measures t/√2 across, so its per-axis tolerance is ±t/(2√2). The circle’s area is πt²/4 against the square’s t²/2 — a ratio of π/2, or about 57.1% more usable area for the same worst-case deviation. Every part in the corner region passes position and fails coordinate tolerancing, which is the entire practical argument for the round zone.

Position tolerance zones and their equivalent coordinate tolerances
Position Ø	Equivalent ± per axis	Max radial deviation	Extra usable area
Ø0.100	±0.0354	0.0500	+57.1%
Ø0.250	±0.0884	0.1250	+57.1%
Ø0.500	±0.1768	0.2500	+57.1%
Ø1.000	±0.3536	0.5000	+57.1%

Units are whatever you are working in — the relationship is dimensionless. Max radial deviation is half the position value, which is the number a CMM report often gives and the number people mistake for the position itself.

The GD&T Material Condition Modifiers This Calculator Applies

Ⓜ Maximum Material Condition (MMC): States that the stated tolerance applies when the feature contains the most material — the largest pin or the smallest hole. As the feature departs from that limit the difference becomes bonus tolerance and is added to the geometric tolerance, which is what makes a functional gauge a valid acceptance method.
Ⓛ Least Material Condition (LMC): The mirror of MMC: the stated tolerance applies when the feature contains the least material — the smallest pin or the largest hole. It is the right modifier when the design risk is thin wall or minimum edge distance rather than assembly clearance.
Ⓢ Regardless of Feature Size (RFS): The default condition since 1994: the geometric tolerance applies at whatever size the feature is actually produced, with no bonus. The circled S is legacy notation retained here because older drawings still carry it; on a current drawing the absence of any material-condition modifier already means RFS.

Sources and Method for the True Position Calculator

GD&T characteristics and modifierscompiled

Unicode Character Database (Unicode License v3); characteristic list cross-checked against NASA MSFC GD&T Basics (public domain); descriptions written from scratch

Verified against gdt: 98 checks passed across 1 harness(es)

Compiled from independent public sources that agree cell-for-cell.

Formula

True position from measured deviation

true position  =  2 × √( Δx² + Δy² )

  Δx, Δy = measured position minus basic (theoretically exact) position

The factor of two makes it a diameter. Position defines a cylindrical zone centred on the true location, so the tolerance and the result are both diametral values.

Bonus tolerance under Ⓜ or Ⓛ

hole at MMC  =  its smallest limit        pin at MMC  =  its largest limit
bonus        =  | actual size − stated-condition size |     (never negative)
allowed Ø    =  stated tolerance + bonus

A feature outside its own size limits earns no bonus — it is a size reject before position is evaluated.

Equivalent coordinate tolerance

± per axis  =  Ø ÷ (2 × √2)
area ratio  =  (π · Ø² ÷ 4) ÷ (Ø² ÷ 2)  =  π ÷ 2  ≈  1.5708

The largest square inscribed in the round zone, and how much smaller it is. This is where the '57% more tolerance' figure comes from.

Assumptions

Δx and Δy are deviations from the BASIC dimensions, measured in the datum reference frame the feature control frame calls out. Establishing that frame correctly is a measurement question this calculator cannot check.
The calculation is two-axis. A true position with three basic dimensions extends the same way — 2 × √(Δx² + Δy² + Δz²) — but a three-axis case usually involves a projected tolerance zone or a feature normal to a compound datum, which is not the same arithmetic.
Everything is unit-agnostic and nothing is converted. Mixing millimetre deviations with an inch tolerance would give a plausible wrong answer, so the calculator never converts and the unit control only labels the output.
Bonus tolerance is applied only when a material-condition modifier is selected. Since 1994 the default with no modifier is RFS: the tolerance applies at whatever size the feature was produced, with no bonus.
Datum shift — the additional tolerance available when a datum feature of size is referenced at MMB — is a separate allowance and is NOT included here.
No text from ASME Y14.5 or ISO 1101 is reproduced anywhere on this page. The standards are cited by number; every description is original prose written for this project.
GD&T characteristics and modifiers scope: EXCLUDED: all ASME Y14.5 definitional text, worked examples and figures. Every description on this site is original prose.

Data last verified

August 1, 2026

How this data is built and checked →

How to Cite This True Position Calculator

Citation

ShopMath. "True Position Calculator — GD&T Position With MMC Bonus." ShopMath, verified August 1, 2026, https://shopmath.org/calculators/true-position

BibTeX

@misc{shopmath-calculators-true-position,
  title        = {True Position Calculator — GD&T Position With MMC Bonus},
  author       = {{ShopMath}},
  year         = {2026},
  howpublished = {\url{https://shopmath.org/calculators/true-position}},
  note         = {Data verified 2026-08-01}
}

Permanent URL

https://shopmath.org/calculators/true-position

Common Questions About GD&T True Position

What is the true position formula?: True position = 2 × √(Δx² + Δy²), where Δx and Δy are the measured deviations from the basic location in the datum reference frame. The factor of two is essential: position defines a diametral tolerance zone, so a feature 0.25 mm from true position consumes a Ø0.50 mm zone. Reporting the radial distance instead of the diameter is the most common mistake in position reporting.
How do you calculate bonus tolerance at MMC?: Bonus is the departure of the feature from its maximum material condition. For a hole, MMC is the smallest allowed size, so bonus = actual size − MMC size; for a pin, MMC is the largest allowed size, so bonus = MMC size − actual size. Add that to the stated tolerance to get the allowed zone. Bonus is never negative, and a feature outside its own size limits earns none — it is a size reject before position is assessed.
Why is a GD&T position tolerance round instead of a square ± box?: Because a round zone gives about 57% more usable area for the same worst-case deviation. The largest square that fits inside a Ø t circle has a per-axis tolerance of ±t/(2√2), and the circle's area exceeds the square's by a factor of π/2. Features that land in the corner region pass position while failing an equivalent coordinate callout — and they still assemble, which is what the tolerance is protecting.
What is the equivalent ± coordinate tolerance for a Ø0.5 position tolerance?: ±0.1768 per axis. That is the largest square zone that fits entirely inside the Ø0.5 circle, so a part meeting it always meets the position callout — but not the other way round, since the circle also accepts everything in the four corner regions the square excludes.
Does this true position calculator handle MMC and LMC?: Yes. Select Ⓜ or Ⓛ, say whether the feature is a hole or a pin, and enter the actual size with both size limits. The calculator works out which limit is the stated material condition, computes the departure as bonus tolerance, adds it to the stated tolerance, and gives the pass or fail against the enlarged zone. Datum shift at MMB is a separate allowance and is not included.