Circumcircle, apothem, central triangle, and diagonal structureLive current inputs
Math & Statistics
Polygon Solver Calculator
Solve a regular polygon from side count and side length, including perimeter, apothem, circumradius, area, interior and exterior angles, diagonals, and a triangulated geometric proof.
Perimeter-
Area-
Apothem-
Circumradius-
Each interior angle-
Each exterior angle-
Total diagonals-
Center triangles-
REGULAR-POLYGON BLUEPRINT
Circumcircle, apothem, central triangle, and diagonal structure
The scaled construction labels one side, one radius, one apothem, and the central angle while a compact inset counts non-adjacent vertex connections.
GEOMETRY RECONCILIATION
One polygon solved through perimeter, triangles, and angle sums
Independent formulas meet at the same area and preserve the distinction between center triangles and vertex diagonals.
| Quantity | Formula | Substitution | Value | Independent check |
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POLYGON ENTRY
Confirm regularity before applying one-side formulas
- Enter an integer side count of at least three.
- Enter one side length in the desired unit.
- Verify the real polygon has equal sides and equal angles.
- Use apothem for center-to-edge distance.
- Use circumradius for center-to-vertex distance.
WHY REGULARITY MATTERS
One side determines the whole polygon only when all sides and angles match
For a regular polygon, the central angle is 360°/n and the center triangles are congruent. This symmetry supports the compact formulas.
An irregular polygon with the same side count and side length can have a different area, radius, and angle structure; coordinates or additional dimensions are then required.
REGULAR-POLYGON SOLUTION
Decompose the polygon into congruent isosceles triangles
Joining the center to every vertex creates n congruent triangles. Half of one triangle gives the tangent relation for apothem; its area then scales by n.
Detailed calculation process and general formulas
P=nsa=s/[2tan(π/n)]R=s/[2sin(π/n)]K=Pa/2D=n(n-3)/2Symbols, meanings, and units
- n
- number of polygon sidescount
- s
- one side lengthlength
- P
- polygon perimeterlength
- a
- apothem from center to side midpointlength
- R
- circumradius from center to vertexlength
- K
- polygon arealength²
- D
- number of distinct diagonalscount
DESIGN DIMENSIONS
Different radii answer different clearance questions
The blueprint distinguishes dimensions often mixed up in specifications.
Across flats
-Twice the apothem gives the distance between opposite supporting lines when n is even.
Across vertices
-Twice the circumradius gives the circumscribed diameter.
Connection count
-The diagonal formula counts all non-edge vertex pairs once.
Decision takeaway: Specify whether a clearance is controlled by flats, vertices, or side length; they are not interchangeable.
APPLICATION BOUNDARIES
Check these before using the solved dimensions
- Regular, not irregular
- Nominal side length
- Corner radius excluded
- No wall thickness
- Planar geometry
- Consistent units
Applied decisions
Two regular-polygon decisions
Hexagonal paving unit
A regular unit is specified by edge length.
What the result clarifies: Area supports coverage while apothem controls row spacing.
Polygonal tank footprint
A regular shell needs outer-vertex and across-flat clearances.
What the result clarifies: Circumradius and apothem answer different site-fit checks.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
The formulas require an ideal regular polygon in a plane. Corner rounding, wall thickness, bevels, joint gaps, irregular fabrication, and three-dimensional effects are outside this model.
Regular Polygon Solver Calculator | Area, Apothem, Radius, Angles, and Diagonals FAQ
What is the difference between apothem and circumradius?
The apothem reaches a side midpoint; the circumradius reaches a vertex.
Why must side count be an integer?
A polygon consists of a whole number of straight sides.
How are diagonals counted?
Each vertex connects to n-3 nonadjacent vertices, then division by two removes double counting.
Can this solve an irregular polygon?
No. Irregular polygons require coordinates, angles, or triangulation data.