Nested polygon geometry beside a log-scale error corridorLive current inputs
Math & Statistics
Circle Approximation Calculator
Approximate π, circumference, and circle area with regular inscribed and circumscribed polygons, track rigorous lower and upper bounds, and visualize convergence as side count increases.
Inscribed circumference bound-
Circumscribed circumference bound-
Inscribed area bound-
Circumscribed area bound-
Lower π estimate-
Upper π estimate-
Relative circumference gap-
Selected side count-
ARCHIMEDEAN CONVERGENCE LAB
Nested polygon geometry beside a log-scale error corridor
The left panel overlays the selected inscribed and circumscribed polygons on the circle. The right panel shows their π bounds squeezing toward π as side count doubles.
BOUND LADDER
How polygon bounds tighten as side count doubles
Every row preserves a lower and upper estimate, so convergence is auditable without relying on the known value of π alone.
| Sides | π lower | π upper | Bound width | Area lower | Area upper | Improvement |
|---|
APPROXIMATION SETUP
Select a side count that exposes convergence rather than hiding it
- Enter a positive radius.
- Choose at least three polygon sides.
- Set a larger maximum side count for the convergence ladder.
- Compare lower and upper bounds together.
- Use the relative gap to decide whether the approximation is adequate.
WHY TWO POLYGONS MATTER
A single approximation has error; paired bounds certify its direction
The inscribed perimeter uses chords and must be shorter than the circumference. The circumscribed perimeter uses tangent-side segments and must be longer.
The radius cancels from the π estimates, but it remains in physical circumference and area bounds. Doubling n rapidly tightens both.
REGULAR-POLYGON BOUNDS
Use chord and tangent geometry to bracket the circle
An inscribed polygon lies inside the circle and underestimates circumference and area. A circumscribed polygon lies outside and overestimates them. Increasing n squeezes both bounds.
Detailed calculation process and general formulas
C₋=2nr sin(π/n)C₊=2nr tan(π/n)A₋=(n/2)r² sin(2π/n)A₊=nr² tan(π/n)C₋/(2r) < π < C₊/(2r)Symbols, meanings, and units
- n
- regular polygon side countcount
- r
- circle radiuslength
- C₋,C₊
- circumference lower and upper boundslength
- A₋,A₊
- area lower and upper boundslength²
- π₋,π₊
- polygon-based bounds on πdimensionless
ERROR CONTROL
Convergence is visible in geometry and numbers
The page does more than draw a many-sided shape.
Guaranteed bracket
-Lower and upper estimates enclose the true circle measure.
Resolution cost
-Side count records how much geometric complexity was used.
Stopping rule
-Relative gap quantifies whether further refinement matters.
Decision takeaway: Report the bound width with the estimate; many sides do not automatically mean enough accuracy for every application.
Applied decisions
Two reasons to approximate a circle with polygons
Historical π construction
Successive regular polygons reproduce the bounding logic used before modern numerical methods.
What the result clarifies: The lower and upper estimates show both convergence and proof direction.
Faceted fabrication
A circular boundary is built from straight segments.
What the result clarifies: The selected n reveals the perimeter and area consequences of faceting.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
These are ideal regular-polygon bounds in exact Euclidean geometry. Fabrication chord tolerances, joint gaps, finite thickness, and numeric rounding can dominate the theoretical approximation at high side counts.
Circle Approximation Calculator | Inscribed and Circumscribed Polygon Bounds FAQ
Why is the inscribed value always lower?
Its edges are chords lying inside the circular arc.
Why is the circumscribed value higher?
Its tangent edges lie outside the circle.
Does radius affect the π estimate?
No. Radius cancels when a perimeter bound is divided by 2r.
How many sides are enough?
Use the displayed relative bound gap and the tolerance of your application.