True Position Formula Calculator

This calculator computes the true position of a feature relative to its datum reference frame, using the standard GD&T formula. You enter the measured X and Y deviations from the nominal location, and the calculator returns the true position diameter (the radial error doubled) and the radial deviation. The result is a single value that you compare against the tolerance stated in your drawing.

Formula

The true position formula per ASME Y14.5 (and ISO 1101) is:

True Position (diameter) = 2 × √(x² + y²)

Where:

• x = measured deviation in the X direction from the true (nominal) position, in mm or inches
• y = measured deviation in the Y direction from the true position, in the same unit

The result is a diameter value. If the drawing calls out a positional tolerance of Ø 0.5 mm, the calculated true position must be ≤ 0.5 mm for the part to be accepted. If the drawing uses a radius (rare), you would compare the radial deviation (√(x²+y²)) directly.

Worked Example

Using the default values from the calculator: x = 0.2 mm, y = 0.3 mm.

1. Square each deviation: 0.2² = 0.04, 0.3² = 0.09
2. Sum the squares: 0.04 + 0.09 = 0.13
3. Take the square root: √0.13 ≈ 0.3606 mm (radial deviation)
4. Double it for the diameter: 2 × 0.3606 = 0.721 mm

The calculator shows 0.721 mm as the true position diameter, and 0.361 mm as the radial deviation. If the drawing tolerance is Ø 0.5 mm, this feature fails because 0.721 > 0.5.

How True Position Differs from Simple Deviation

True position is often misunderstood as just the distance from nominal in one axis. In reality, it accounts for the combined offset in all axes (typically X and Y for a 2D feature, or X, Y, and Z for a 3D feature). The formula above is for the 2D case, which is the most common for holes and pins. For a 3D feature, the formula becomes 2 × √(x² + y² + z²). The result is always a diameter, not a radius, because the tolerance zone is a cylinder (or sphere) centered on the true position.

When to Use This Calculator

Use this calculator when you have measured coordinates from a CMM (coordinate measuring machine) or a manual inspection setup. You need to know the deviation of the actual feature location from the basic (nominal) dimensions shown on the drawing. The deviations are not the coordinates themselves; they are the differences between measured and nominal values. For example, if the nominal X is 25.00 mm and the measured X is 25.13 mm, the deviation x = 0.13 mm.

This calculator is for the simplest case: a single feature with no material condition modifiers (like MMC or LMC). If the drawing includes a maximum material condition (MMC) symbol, the tolerance gets a bonus equal to the amount the feature departs from MMC. That bonus is added to the tolerance before comparison, but the true position formula itself remains the same.

Common Mistakes and Pitfalls

• Forgetting to double the radial deviation: The true position tolerance is always a diameter, so the radial distance must be multiplied by 2. Many errors come from comparing the radius to the diameter tolerance.
• Using coordinates instead of deviations: If you enter the absolute measured coordinates (e.g., 25.13) instead of the difference from nominal (0.13), the result will be wildly wrong.
• Mixing units: Ensure both deviations are in the same unit (mm, inches, etc.). The calculator assumes you enter them in the same unit; it does not convert.
• Ignoring the third axis: For features where axial location matters (like a counterbore depth), you must include the Z deviation. The 2D formula would underestimate the true position.
• Misreading the tolerance: The drawing might specify a positional tolerance of 0.5 mm diameter, not radius. Always check the leader line and any Ø symbol.

Limitations of the Formula

The true position formula only accounts for the location of a single point (usually the center of a hole or pin). It does not account for form errors like circularity or straightness, nor does it account for orientation errors like perpendicularity. Those are separate GD&T controls. Also, the formula assumes the deviations are measured relative to the correct datum reference frame. If the datum features are not set up correctly, the measured deviations will be meaningless.

For a pattern of holes, true position is evaluated per hole, not as a group. Some drawings use composite positional tolerances (two lines in the feature control frame) – the upper segment controls location relative to datums, the lower segment controls orientation only. This calculator handles only the single-segment case.

Frequently Asked Questions

What does the true position result mean?

The result is the diameter of the smallest cylinder (or circle) centered on the true position that contains the measured center of the feature. If that diameter is less than or equal to the tolerance in the feature control frame, the part passes. If it is larger, the feature is out of tolerance.

Can I use this calculator for a 3D true position?

No, this calculator is for 2D (X and Y) only. For a 3D true position, you would need to include the Z deviation in the formula: 2 × √(x² + y² + z²). You can still use this calculator by entering the combined radial deviation from X and Y as one input and Z as the other, but that would be an approximation only if you treat the 3D case as a 2D projection, which is not correct. Better to use a dedicated 3D calculator or add the Z term manually.

What is the difference between true position and concentricity?

True position controls the location of a feature’s center point relative to a datum. Concentricity controls the location of the median points of all opposed elements of a surface, which is a much stricter and harder-to-measure control. In practice, true position is far more common and easier to verify with a CMM.

How do I handle MMC (maximum material condition) with this calculator?

When MMC is specified, the tolerance is increased by the amount the feature departs from its maximum material size. For example, if a hole has a positional tolerance of Ø 0.5 mm at MMC and the hole is actually 0.2 mm larger than the MMC size, the tolerance becomes 0.5 + 0.2 = 0.7 mm. You would then compare your calculated true position against this adjusted tolerance. The formula itself remains unchanged.

Can true position be negative?

No, true position is always a positive value because it is a distance (diameter). If you get a negative result, you likely entered a negative deviation and the calculator handled it correctly (since squaring removes the sign), but the output is always positive.

What if my drawing uses inches instead of millimeters?

The calculator works with any unit as long as you enter both deviations in the same unit. The result will be in that same unit. For example, if you enter x = 0.008 in and y = 0.012 in, the true position will be in inches. Just be consistent.

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in x | Measured X deviation | mm | 0.2 | 0
in y | Measured Y deviation | mm | 0.3 | 0
out tp | True position (diameter) | mm | 2 * sqrt(x^2 + y^2) | 3
out radial | Radial deviation | mm | sqrt(x^2 + y^2) | 3