True Position Calculation: Formula and Examples

True position is a geometric dimensioning and tolerancing (GD&T) control that describes how far a measured feature’s actual location may be from its theoretically exact location. For a circular feature, calculate the diametrical true-position error as 2 × √(ΔX² + ΔY²), where ΔX and ΔY are the measured offsets from the intended X and Y coordinates; the result is compared with the stated position tolerance.

Formula

For a circular feature located by two perpendicular coordinate measurements, the diametrical true-position error is:

TP = 2 × √(ΔX² + ΔY²)

• TP is the diametrical true-position error, in the same length unit as the offsets.
• ΔX is the actual measured X coordinate minus the theoretically exact X coordinate.
• ΔY is the actual measured Y coordinate minus the theoretically exact Y coordinate.

The calculator uses millimetres, but the same formula works in inches if both inputs are entered in inches. The radial offset, √(ΔX² + ΔY²), is also shown as a helper value. In GD&T, position tolerance for a circular feature is normally stated as a diameter, so the first output doubles the radial offset. The calculator’s defaults use ΔX = 0.3 mm and ΔY = 0.4 mm.

Worked Example

Using the calculator’s default values:

1. Square each coordinate offset: 0.3² = 0.09 and 0.4² = 0.16.
2. Add the squares: 0.09 + 0.16 = 0.25.
3. Take the square root: √0.25 = 0.5 mm. This is the radial offset.
4. Double it to get diametrical true-position error: 2 × 0.5 = 1.00 mm.

The feature’s position error is therefore 1.00 mm diametrically. Whether it passes depends on the applicable tolerance and the drawing’s specified conditions.

What True Position Means

A design drawing gives a feature a theoretically exact location, often established by basic dimensions from datums. The measured feature will not be perfectly located in practice. True position expresses the allowed location variation relative to that theoretical location and the drawing’s datum reference frame.

For a hole, the controlled feature is generally its axis. The position tolerance defines a cylindrical zone around the theoretically exact axis. The actual axis must lie within that zone for the feature to meet the position requirement. For a planar or noncircular feature, the controlled geometry and tolerance-zone shape can differ, so the simple two-coordinate calculation may not capture the full requirement.

True position is not the same as an X or Y coordinate error by itself. It combines perpendicular offsets into one radial distance, then commonly expresses that distance as a diametrical tolerance-zone value.

How to Use the Calculation

Use the calculator for a basic two-dimensional location check when you have measured coordinate offsets for a circular feature. Enter offsets from the nominal location, not the absolute coordinates, and use the same unit for both.

• Find the theoretically exact X and Y coordinates from the drawing.
• Record the measured X and Y coordinates in the same datum setup.
• Subtract nominal from measured coordinates to get ΔX and ΔY. The signs indicate direction, but the squared formula makes the final distance positive.
• Compare the calculated diametrical error with the position tolerance, taking any material-condition modifiers and datum requirements on the drawing into account.

For example, a measured coordinate could be 25.3 mm where the nominal coordinate is 25.0 mm, giving ΔX = +0.3 mm. If the other axis is 40.4 mm against a nominal 40.0 mm, then ΔY = +0.4 mm. The calculation uses those differences, not 25.3 and 40.4 directly.

Interpreting the Tolerance

If a drawing specifies a position tolerance of Ø1.2 mm under the applicable conditions, a calculated error of 1.00 mm is within that numerical limit. A result of 1.30 mm would exceed it. The comparison is valid only if the measurement and tolerance refer to the same feature, datum reference frame, and inspection conditions.

Some drawings attach a material-condition modifier, such as maximum material condition, to a position tolerance. That can change the allowable boundary or provide additional tolerance as the actual feature departs from the specified material condition. Do not assume the printed value is the only allowance: interpret the feature control frame and the governing drawing standard. The calculator does not determine bonus tolerance, datum mobility, or functional-gage acceptance.

Common Errors and Limits

• Doubling incorrectly: The square-root result is radial offset. For the usual diametrical position value, double it. Comparing the undoubled radius directly with a diametrical tolerance can give the wrong conclusion.
• Using absolute coordinates: Enter coordinate deviations from the true location, not the measured coordinate values themselves.
• Mixing units: Do not combine millimetres with inches. Convert both offsets and the tolerance to one unit before comparing.
• Rounding too early: Keep full precision during intermediate steps and round only the reported result. A value near the tolerance limit may need more digits or a defined measurement decision rule.
• Ignoring the datum setup: Offsets measured from a different origin or alignment are not necessarily comparable to the drawing’s datum reference frame.

This formula calculates a two-dimensional location error from two coordinate deviations. It does not account by itself for a feature tilted out of plane, size variation, form error, measurement uncertainty, or a complex datum scheme. For a full GD&T evaluation, inspect the complete feature control frame and use an appropriate measurement method, such as a coordinate-measuring system, aligned to the drawing’s datums.

Frequently Asked Questions

Is true position twice the distance from the nominal location?
For the common circular-feature calculation, yes: the Euclidean distance from the nominal center is the radial offset, and diametrical true-position error is twice that value.

Can true position be negative?
No. X and Y deviations can be positive or negative, but the calculated distance is nonnegative because the deviations are squared before the square root is taken.

What if the drawing gives only one coordinate deviation?
If the other perpendicular deviation is known to be zero, use zero for it. Otherwise, one deviation alone is not enough to calculate the two-dimensional position error.

Does a result below the tolerance always mean the part passes?
Not necessarily. The result must be evaluated against the full drawing requirements, including datums, modifiers, feature size, and any specified inspection or decision rules.

Can I calculate position from three coordinates?
For a three-dimensional point, the Euclidean distance is √(ΔX² + ΔY² + ΔZ²). GD&T acceptance may still involve a specified tolerance zone and datum setup, so a general point-distance calculation is not a complete drawing evaluation.

Related

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• Calculate True Positive and False Positive Rates

in dx | X offset from true position | mm | 0.3 | 0
in dy | Y offset from true position | mm | 0.4 | 0
out tp | True position error | mm | 2 * sqrt(dx^2 + dy^2) | 2
out r | Radial offset | mm | sqrt(dx^2 + dy^2) | 2