One regular polygon unfolded into congruent proof trianglesUpdates with every input
Math & Statistics
Polygon Step-by-Step Calculator
Derive a regular polygon from side count and side length, following the center-triangle construction from central angle through apothem, perimeter, area, interior-angle sum, and diagonals.
Central angle -
Apothem -
Perimeter -
Area -
Each interior angle -
Interior-angle sum -
Circumradius -
Diagonals -
CENTER-TRIANGLE DERIVATION
One regular polygon unfolded into congruent proof triangles
The live construction isolates a half-triangle, labels its opposite and adjacent legs, and connects every formula to the same center fan.
DERIVATION LEDGER
Every result in dependency order
Each row uses a previously established quantity, making the chain auditable instead of presenting disconnected answers.
| Step | Quantity | Formula | Live substitution | Result |
|---|
GEOMETRY ENTRY
Confirm regularity before using a one-side derivation
- Enter a whole side count of at least three.
- Measure one representative side in a consistent unit.
- Confirm the polygon is equilateral and equiangular.
- Read apothem for center-to-edge clearance and radius for center-to-vertex clearance.
- Use the ledger to retain rounding until the final reported value.
WHY THE FAN WORKS
Symmetry turns a many-sided figure into one repeatable right triangle
The center fan is valid because equal central angles and equal radii create congruent center triangles. That same symmetry makes every side share one apothem.
Area can be reconciled two ways: n times one center-triangle area, or one-half perimeter times apothem. Their agreement is the primary check.
Subject fundamentals
Five ideas that control this calculation
Regularity is an assumption
One side determines the rest only when every side and central angle is equal. An irregular polygon needs coordinates or separate measurements.
The center fan is the starting model
Joining the center to every vertex creates n congruent isosceles triangles and exposes the central angle 360 degrees divided by n.
The apothem is perpendicular
Bisecting one center triangle creates a right triangle whose short leg is half the side and whose other leg is the apothem.
Perimeter and area use different dimensions
Perimeter totals edge length; area combines perimeter with the inward distance to the center. They must not be compared as interchangeable quantities.
Rounding belongs at the end
Trigonometric intermediates retain full precision so the area and center-triangle reconciliation do not drift because of an early rounded apothem.
TRIGONOMETRIC CONSTRUCTION
Bisect one center triangle before computing the whole polygon
Joining the center to each vertex creates n congruent isosceles triangles. Bisecting one gives a right triangle with opposite leg s/2 and angle pi/n.
Detailed calculation process and general formulas
alpha = pi / na = s / [2 tan(alpha)]R = s / [2 sin(alpha)]P = nsK = Pa / 2D = n(n - 3) / 2Symbols, meanings, and units
- n
- number of equal sidescount
- s
- one side lengthlength
- alpha
- half of one central angleradians
- a
- apothemlength
- R
- circumradiuslength
- P
- perimeterlength
- K
- arealength squared
- D
- distinct vertex diagonalscount
DESIGN INTERPRETATION
Three dimensions answer three different fabrication questions
A side callout alone does not describe the full envelope.
Edge material
-Perimeter prices trim, fencing, or seam length.
Across-flat clearance
-Twice the apothem controls spacing between opposite support lines.
Outer envelope
-Twice the circumradius controls the smallest centered circle containing every vertex.
Decision takeaway: State whether a specification is side length, across flats, or across vertices.
FIELD CHECKS
Conditions the formulas cannot verify
- Equal side lengths
- Equal interior angles
- Planar figure
- Sharp theoretical corners
- No joint allowance
- One consistent length unit
Applied decisions
Where the derivation changes a real decision
Octagonal display plinth
A fabricator knows the edge length but needs sheet area and the outer turning envelope.
What the result clarifies: Area prices the face while radius controls clearance.
Regular garden border
A layout uses twelve equal panels around a center point.
What the result clarifies: The apothem locates each panel line and the perimeter totals border material.
Worked default scenario
Current-input substitution and reconciliation
Key terms
Glossary for interpreting the result
- Regular polygon
- A polygon with equal sides and equal interior angles.
- Central angle
- The angle at the center between radii to adjacent vertices.
- Apothem
- The perpendicular distance from the center to a side.
- Circumradius
- The distance from the center to any vertex.
- Interior-angle sum
- The total of all interior angles, equal to 180(n-2) degrees.
- Diagonal
- A segment joining two nonadjacent vertices.
Method references
References for this calculator's specific method
Scope and limitations
This model applies only to ideal regular, planar polygons. Irregular sides, rounded corners, wall thickness, kerf, joint gaps, and three-dimensional construction require additional geometry.
Polygon Step-by-Step Calculator | Regular Polygon Derivation FAQ
Why is the half-angle pi divided by n?
A full turn is 2pi; one center triangle uses 2pi/n and its right-triangle half uses pi/n.
Why does area equal Pa/2?
Each center triangle has base s and height a; summing ns times a divided by two gives Pa/2.
Are apothem and radius interchangeable?
No. The apothem ends at a side midpoint; the circumradius ends at a vertex.
Can I use a decimal side count?
No. A polygon has a whole number of sides.