Radius density mapped into circumference and right-skewed areaLive current inputs
Math & Statistics
Circle Distribution Calculator
Transform a normally uncertain radius into circumference and area distributions, preserve the exact nonlinear area moments, and show why symmetric radius uncertainty creates skewed area outcomes.
Mean circumference-
Circumference standard deviation-
Exact mean area-
Exact area standard deviation-
Radius lower quantile-
Radius upper quantile-
Transformed area interval-
Area coefficient of variation-
NONLINEAR TRANSFORMATION
Radius density mapped into circumference and right-skewed area
Aligned probability markers show the same underlying radius quantiles on three scales, while the area panel widens because squaring amplifies high-radius outcomes.
QUANTILE TRANSFORM
One probability level expressed as radius, circumference, and area
Because both transforms are monotonic for positive radius, matched quantiles can be converted directly.
| Probability | Radius | Circumference | Area | Distance from mean area | Interpretation |
|---|
UNCERTAINTY ENTRY
Describe radius measurement uncertainty before transforming it
- Enter the mean radius.
- Enter one standard uncertainty, not a full tolerance width.
- Choose the central probability coverage.
- Compare πμ² with the exact mean area.
- Check whether negative-radius probability is negligible.
WHY MEAN AREA IS LARGER
Squaring rewards high deviations more than it penalizes equal low deviations
For a normal radius, E[R²]=μ²+σ². The variance term is why the exact expected area exceeds the area calculated from the mean radius alone.
The model is sensible only when μ is several standard deviations above zero. If negative-radius probability is material, use a bounded positive distribution rather than an untruncated normal.
MOMENT TRANSFORMATION
Use linear moments for circumference and exact squared-normal moments for area
Circumference is a linear transform of radius. Area is proportional to R², so its expected value includes both mean² and variance; simply squaring the mean radius understates expected area.
Detailed calculation process and general formulas
C=2πRE[C]=2πμE[A]=π(μ²+σ²)Var(A)=π²(2σ⁴+4μ²σ²)Aq=πRq²Symbols, meanings, and units
- R
- uncertain radiuslength
- μ
- mean radiuslength
- σ
- radius standard uncertaintylength
- C
- circumferencelength
- A
- circle arealength²
- Rq,Aq
- matched radius and area quantileslength; length²
TRANSFORMATION EFFECTS
Linear and nonlinear outputs carry uncertainty differently
Matched quantiles make the contrast visible.
Linear scaling
-Circumference keeps the radius distribution's standardized shape.
Curvature bias
-Exact mean area includes the radius variance contribution.
Relative spread
-Area coefficient of variation expresses amplified uncertainty.
Decision takeaway: Do not calculate an expected area by squaring only the expected radius when radius uncertainty is material.
DISTRIBUTION ASSUMPTIONS
When the normal-radius model needs replacement
- Large probability below zero
- Asymmetric tolerance
- Calibration bias
- Radius varies spatially
- Repeated measurements correlated
- Diameter measured instead of radius
Applied decisions
Two transformations of radius uncertainty
Measured circular plate
Repeated radius measurements produce a mean and standard uncertainty.
What the result clarifies: The expected material area includes the variance term and the interval is asymmetric after squaring.
Circular coverage estimate
A nominal reach radius varies from trial to trial.
What the result clarifies: Area variability is proportionally larger than radius variability.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
This model assumes an untruncated normal radius and a perfect circle. It is unsuitable when negative radii have non-negligible probability, when roundness error matters, or when uncertainty is better described by bounded, asymmetric, or correlated measurements.
Circle Distribution Calculator | Radius Uncertainty to Circumference and Area FAQ
Why is mean area not π times mean radius squared?
Because E[R²]=E[R]²+Var(R), so radius variance contributes to expected area.
Is circumference also skewed?
No. A linear transform of a normal radius remains normal.
Why can the area interval be asymmetric?
Area squares the radius quantiles, stretching the upper side more strongly.
What if I measured diameter?
Convert both the diameter mean and its standard uncertainty by dividing by two.