Inscribed and circumscribed polygons with a logarithmic error lensUpdates with every input
Math & Statistics
Polygon Approximation Calculator
Bracket circle circumference and area with inscribed and circumscribed regular polygons, quantify absolute and relative error, and watch the bounds contract as the side count doubles.
Lower pi bound -
Upper pi bound -
Pi interval width -
Lower circumference -
Upper circumference -
Lower area -
Upper area -
Relative bound width -
ARCHIMEDEAN BOUNDS
Inscribed and circumscribed polygons with a logarithmic error lens
The left construction shows the geometric bracket; the right curve shows how the upper-lower gap contracts under repeated side doubling.
BOUND CONTRACTION
Circumference and area brackets at every doubling level
Each row is a valid lower and upper bound, not a rounded equality.
| Sides | Lower pi | Upper pi | Pi width | Circumference bracket | Area bracket | Tightening |
|---|
APPROXIMATION SETUP
Choose side count for accuracy, not appearance alone
- Enter the physical circle radius.
- Choose at least three polygon sides.
- Set how many doubling checkpoints to display.
- Read lower and upper values as a bracket.
- Increase n until the interval meets the required precision.
WHY THE TRUE VALUE IS TRAPPED
Chord paths cut corners while tangent paths travel outside
Every inscribed edge is a chord and lies inside its corresponding arc. Every circumscribed edge is tangent-based and lies outside the arc.
Doubling n reduces the angular span per edge. Both bounds approach the same limit, giving a geometric convergence certificate.
Subject fundamentals
Five ideas that control this calculation
The lower path is inscribed
Chord edges lie inside the circle, making their total perimeter and enclosed area genuine lower bounds.
The upper path is circumscribed
Tangent edges meet outside the circle, so their perimeter and area remain upper bounds for the corresponding circle quantities.
Pi emerges after radius cancellation
Dividing the perimeter bounds by the diameter removes the chosen radius and leaves n sin(pi/n) below pi and n tan(pi/n) above it.
Doubling tightens one certified interval
Each level halves the central angle and narrows the lower-to-upper gap without pretending that either endpoint is the exact value.
Area and circumference share geometry
The same side count brackets both quantities, but the displayed dimensions remain distinct: length for circumference and square units for area.
GEOMETRIC BRACKETING
Use chords inside the circle and tangents outside it
An inscribed n-gon has shorter perimeter than the circle; a circumscribed n-gon has longer perimeter. Dividing both by diameter brackets pi.
Detailed calculation process and general formulas
pi_low = n sin(pi/n)pi_high = n tan(pi/n)C_low = 2nR sin(pi/n)C_high = 2nR tan(pi/n)A_low = nR^2 sin(2pi/n)/2A_high = nR^2 tan(pi/n)Symbols, meanings, and units
- n
- polygon side countcount
- R
- circle radiuslength
- pi_low
- inscribed-perimeter pi bounddimensionless
- pi_high
- circumscribed-perimeter pi bounddimensionless
- C_low,C_high
- circumference bracketlength
- A_low,A_high
- area bracketlength squared
PRECISION CONTROL
Report an interval before choosing a rounded estimate
Bounds communicate more than a single decimal.
Guaranteed lower value
-The inscribed polygon cannot exceed the ideal circle measure.
Guaranteed upper value
-The circumscribed polygon cannot fall below it.
Resolution
-Interval width states the remaining geometric uncertainty.
Decision takeaway: A displayed decimal is justified only when the bound width is narrower than the rounding place.
NUMERICAL CARE
Keep these distinctions explicit
- Lower bound versus estimate
- Upper bound versus estimate
- Radians inside trig functions
- Side count before rounding
- Physical units versus pi bounds
- Selected n versus doubling sequence
Applied decisions
Approximation questions answered by the bounds
Segmented circular curb
Straight modules approximate a circular border.
What the result clarifies: The circumference bracket quantifies the difference between chord material and tangent layout.
Teaching pi geometrically
A class doubles polygon sides from a hexagon onward.
What the result clarifies: The error lens makes convergence visible without assuming pi first.
Worked default scenario
Current-input substitution and reconciliation
Key terms
Glossary for interpreting the result
- Lower bound
- A certified value that does not exceed the true circle quantity.
- Upper bound
- A certified value that is not below the true circle quantity.
- Chord
- A straight segment joining two points on a circle.
- Tangent
- A line touching a circle at one point and perpendicular to the radius there.
- Interval width
- Upper bound minus lower bound, used as the convergence measure.
- Doubling level
- One step that replaces n sides with 2n sides.
Method references
References for this calculator's specific method
Scope and limitations
Floating-point trigonometry is used to evaluate the theoretical bounds. At very high side counts, machine precision limits further numerical tightening even though the mathematical sequence continues to converge.
Polygon Approximation Calculator | Pi Bounds and Circle Error FAQ
Why is the lower pi bound n sin(pi/n)?
It is the inscribed n-gon perimeter divided by the circle diameter.
Why is tangent used for the upper bound?
A tangent-sided polygon encloses the circle and therefore has a longer perimeter.
Does radius affect the pi bounds?
No. Radius scales circumference and area but cancels from the dimensionless pi bounds.
Why double the side count?
Doubling gives a clear nested refinement and rapidly contracts the bound interval.