One side-length bell transformed into perimeter and area densitiesUpdates with every input
Math & Statistics
Polygon Distribution Calculator
Propagate a measured regular-polygon side-length distribution into perimeter, apothem, and area distributions, including exact mean-area bias from squaring and a live percentile density.
Mean perimeter -
Perimeter SD -
Expected area -
Area SD -
Median-model area -
Squaring bias -
Central area band -
Area coefficient of variation -
NONLINEAR OUTPUT DISTRIBUTION
One side-length bell transformed into perimeter and area densities
Perimeter remains linear and symmetric; area follows s squared and becomes right-skewed, with the chosen central interval shaded.
PERCENTILE TRANSFORM
Probability checkpoints mapped through polygon geometry
Rows use normal side-length quantiles and transform each quantile through the exact regular-polygon formulas.
| Percentile | Side length | Perimeter | Apothem | Area | Band status |
|---|
UNCERTAINTY ENTRY
Use a standard deviation, not a plus-minus tolerance
- Enter the regular side count.
- Enter the best estimate of side length.
- Convert measurement evidence to a standard deviation.
- Choose a central probability coverage.
- Interpret percentiles under the stated normal model.
WHY EXPECTED AREA EXCEEDS AREA AT THE MEAN
Squaring preserves positive spread as an added variance term
For K=cs squared, E[K]=cE[s squared]=c(mu squared plus sigma squared). Evaluating only K(mu) omits c sigma squared.
A normal model assigns tiny probability to negative sides. When sigma is large relative to mu, a positive or bounded measurement model is more appropriate.
Subject fundamentals
Five ideas that control this calculation
The model treats side length as random
Mean and standard deviation describe repeated realizations of one common side-length process, not different deterministic sides in one polygon.
Perimeter is a linear transformation
Multiplying side length by n multiplies both its mean and its standard deviation by n.
Area is a nonlinear transformation
Because regular-polygon area is proportional to side length squared, its expected value includes both squared mean and variance.
Coverage starts in side-length space
A normal quantile creates the entered central coverage band before the monotonically increasing area equation maps its endpoints.
Correlation is outside this model
The calculation assumes one common random side-length driver. Separate correlated side measurements require a covariance model rather than this scalar transformation.
UNCERTAINTY TRANSFORMATION
Propagate the measured side before summarizing nonlinear area
Perimeter is a linear multiple of side length. Area equals c times s squared, so its expected value includes the variance term rather than just the square of the mean.
Detailed calculation process and general formulas
c = n / [4 tan(pi/n)]P = nsK = cs^2E[K] = c(mu^2 + sigma^2)Var(K) = c^2(2sigma^4 + 4mu^2 sigma^2)Symbols, meanings, and units
- mu
- mean side lengthlength
- sigma
- side-length standard deviationlength
- c
- regular-polygon area coefficientdimensionless
- P
- perimeterlength
- K
- arealength squared
- E[K]
- expected area under the entered modellength squared
DISTRIBUTION READOUT
Keep location, spread, and model bias separate
Each statistic answers a different reporting question.
Expected output
-Mean area includes the nonlinear variance correction.
Central range
-The percentile band communicates modeled output spread.
Relative variability
-The coefficient of variation scales area spread to its mean.
Decision takeaway: Do not describe a percentile band as a manufacturing tolerance unless the measurement model supports that interpretation.
MODEL EVIDENCE
Record what supports the entered standard deviation
- Instrument resolution
- Repeat measurements
- Calibration uncertainty
- Operator variation
- Temperature effects
- Correlation across sides
Applied decisions
When a distribution adds information beyond one nominal area
Batch-cut regular panels
Repeated edge measurements show small process variation.
What the result clarifies: The area band converts edge variability into expected material coverage variability.
Survey-derived regular footprint
A nominal side comes from repeated observations.
What the result clarifies: The bias term prevents area-at-the-mean from being mislabeled as expected area.
Worked default scenario
Current-input substitution and reconciliation
Key terms
Glossary for interpreting the result
- Mean side length
- The expected center of the assumed side-length distribution.
- Standard deviation
- The typical spread of side length around its mean.
- Coverage band
- A central probability interval under the stated normal model.
- Nonlinear bias
- The difference E[K]-K(E[s]) created by squaring a variable input.
- Coefficient of variation
- Standard deviation divided by mean, reported here for area.
- Propagation of uncertainty
- The transformation of input variability into output variability through an equation.
Method references
References for this calculator's specific method
Scope and limitations
This model assumes one normally distributed common side dimension and an ideal regular polygon. It does not model independent unequal sides, correlations, systematic calibration bias, truncation, or irregular geometry.
Polygon Distribution Calculator | Side-Uncertainty Area Distribution FAQ
Why is area skewed when side length is normal?
Area depends on the square of side length, a nonlinear transformation.
Why is perimeter still symmetric?
Perimeter is a positive constant times side length under this model.
Can I enter a maximum tolerance as SD?
Not directly. Convert it according to the assumed tolerance distribution and evidence.
What happens when SD is zero?
All distributions collapse to the deterministic regular-polygon values.