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Regular Octagon Area and Three Diagonal Lengths

Regular Octagon Area and Three Diagonal Lengths. 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: Side length (a) 2 gives Perimeter 16, Area 19.313708, Longest diagonal (l) 5.2262519, Medium diagonal (m) 4.8284271, Shortest diagonal (s) 3.6955181, Circumcircle radius (R) 2.6131259, Incircle radius (r) 2.4142136.

A = 2(1+√2)a²; P = 8a; dlong = 2R; dmedium = 2r

Worked example

Enter these known values and leave the other values blank.

Side length (a)
2 cm
Perimeter
16 cm
Area
19.314 cm²
Longest diagonal (l)
5.226 cm
Medium diagonal (m)
4.828 cm
Shortest diagonal (s)
3.6955 cm
Circumcircle radius (R)
2.613 cm
Incircle radius (r)
2.414 cm

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. Three diagonal lengths connect vertices separated by different numbers of edges.
  • 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

Which octagon diagonal passes through its center?

The longest diagonal joins opposite vertices and equals twice the circumradius. The medium diagonal equals twice the apothem.

References

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

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