Area fill and perimeter convergence toward the circumscribed circleUpdates with every input
Math & Statistics
Polygon Value Table Calculator
Compare regular polygons across a controlled side-count sweep while keeping circumradius fixed, revealing how perimeter, area, apothem, and circular fill converge as n increases.
Generated rows -
Selected area -
Circle-area fill -
Circumference shortfall -
Selected side length -
Selected apothem -
Area gain from prior row -
Limiting shape -
SIDE-COUNT RESPONSE
Area fill and perimeter convergence toward the circumscribed circle
Two normalized curves show diminishing returns from adding sides while a live marker identifies the selected polygon.
VALUE TABLE
Regular-polygon geometry at each side-count checkpoint
Circumradius is held constant so changes come only from side count.
| Sides | Side length | Apothem | Perimeter | Area | Circle fill | Perimeter shortfall |
|---|
TABLE DESIGN
Choose a sweep that answers one comparison question
- Enter the common outer radius.
- Set an integer start and end.
- Choose an increment that produces a readable table.
- Set a selected n for the live marker.
- Compare marginal gain, not only final area.
DIMINISHING GEOMETRIC RETURN
Each added side improves circular fill by less than the previous additions
The polygon remains inside its circumcircle, so both area and perimeter approach their circle limits from below.
A large n may offer little geometric gain while increasing joints, vertices, or manufacturing effort. The table exposes that tradeoff.
Subject fundamentals
Five ideas that control this calculation
The circumradius stays fixed
Every row uses the same center-to-vertex distance, making side count the only controlled geometric change in the sweep.
The polygon is inscribed
Its sides are chords inside the circle, so polygon perimeter and area approach but do not exceed the circle limits.
Side length shrinks as n grows
More vertices divide the same circumference into shorter chords even though the overall radius does not change.
Fill and perimeter gap are distinct
Area fill measures occupied surface while perimeter gap measures boundary shortfall; they converge at different rates and answer different design questions.
The selected row is independent
The selected side count is calculated directly even when it is not one of the displayed sweep increments.
CONTROLLED N-SWEEP
Hold the outer radius constant before comparing polygon families
At fixed R, increasing n shortens each edge and pushes the perimeter and area upward toward their circle limits.
Detailed calculation process and general formulas
s_n = 2R sin(pi/n)a_n = R cos(pi/n)P_n = 2nR sin(pi/n)K_n = nR^2 sin(2pi/n) / 2fill_n = K_n / (pi R^2)Symbols, meanings, and units
- n
- side count for one rowcount
- R
- fixed circumradiuslength
- s_n
- edge length at n sideslength
- a_n
- apothem at n sideslength
- P_n
- perimeter at n sideslength
- K_n
- area at n sideslength squared
SELECTION SIGNALS
Use the curve as a design screen, not a claim that more sides are always better
Three outputs summarize different costs of discretizing a circle.
Material utilization
-Circle-area fill measures the occupied share of the outer envelope.
Boundary approximation
-Perimeter shortfall compares the polygon edge total with circumference.
Fabrication complexity
-Row count and side count indicate how many repeated edges or joints are required.
Decision takeaway: Select the smallest side count that satisfies the required fill or boundary accuracy.
COMPARISON CONTROLS
Do not mix these alternative bases
- Fixed circumradius
- Fixed apothem
- Fixed side length
- Fixed perimeter
- Fixed area
- Fixed bounding box
Applied decisions
Decisions clarified by the side-count table
Circular platform approximation
A segmented platform must fit within a fixed circular clearance.
What the result clarifies: The fill curve shows how much floor area each extra panel recovers.
Polygonal duct selection
A fabricator compares joint count with section area at one outer radius.
What the result clarifies: The table identifies the point where added seams produce little area benefit.
Worked default scenario
Current-input substitution and reconciliation
Key terms
Glossary for interpreting the result
- Inscribed polygon
- A polygon whose vertices all lie on a common circle.
- Fixed-radius sweep
- A comparison that varies side count while holding circumradius constant.
- Circle fill
- Polygon area divided by the area of its circumcircle.
- Perimeter gap
- The percentage shortfall from the circle circumference.
- Marginal area gain
- The added area relative to the preceding comparison row.
- Convergence limit
- The circle value approached as the number of sides increases without bound.
Method references
References for this calculator's specific method
Scope and limitations
The comparison holds circumradius constant. Results differ when apothem, side length, perimeter, or area is held constant. Fabrication cost is not inferred from geometry alone.
Polygon Value Table Calculator | Side-Count Response at Fixed Radius FAQ
Why does area rise with side count?
At fixed circumradius, the inscribed polygon fills more of the circle as its edges become shorter.
Why does the curve flatten?
The polygon approaches a finite circle limit, so marginal gains diminish.
Can the selected side count fall outside the table?
Yes. It is still calculated and marked, while the table follows its own sweep.
Is a 100-sided polygon a circle?
No. It is a close polygonal approximation with finite straight edges.