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Regular Polygon Area from Side, Circumradius or Apothem

Regular Polygon Area from Side, Circumradius or Apothem. Use unit controls and documented positive geometric roots to compare base, lateral and total measures.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

Example in initial displayed units: Number of sides 6, Side a 2 gives Area 10.392305, Circumradius 2, Inradius/apothem 1.7320508.

A = na²/(4 tan(π/n)) = nR² sin(2π/n)/2 = nr² tan(π/n)

Worked example

Enter these known values and leave the other values blank.

Number of sides
6
Side a
2 cm
Circumradius
2 cm
Inradius/apothem
1.732 cm
Area
10.392 cm²

Assumptions and limitations

  • The polygon is regular: all sides and all interior angles are equal. Circumradius reaches a vertex and apothem reaches an edge midpoint. These formulas do not apply to an irregular polygon with the same side count. Enter an integer side count of at least three. Drawings for thirteen or more sides are schematic examples rather than an exact outline for every count.
  • Select compatible units and enough independent dimensions. Clear unknown values before solving backward. Numeric precision and display rounding do not quantify uncertainty in measured inputs.
  • Arithmetic uses 64 significant digits. Original reference diagrams explain the dimensions and do not follow entered proportions.

Common questions

Can these area formulas be used for irregular polygons?

No. Equal side count alone is insufficient. These formulas assume equal sides and equal angles, which fix the relation between side, circumradius and apothem.

References

The international acre is exactly 43560 square international feet; Bookify replaces the rounded captured area factor.

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