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Regular Pentagon Area, Diagonal, Height and Apothem

Regular Pentagon Area, Diagonal, Height and 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: Side (a) 2 gives Perimeter 10, Area 6.8819096, Diagonal (d) 3.236068, Height (h) 3.0776835, Circumcircle radius (R) 1.7013016, Incircle radius (apothem) (r) 1.3763819.

A = 5a²/(4 tan(π/5)); P = 5a; d = a(1+√5)/2

Worked example

Enter these known values and leave the other values blank.

Side (a)
2 cm
Perimeter
10 cm
Area
6.882 cm²
Diagonal (d)
3.236 cm
Height (h)
3.0777 cm
Circumcircle radius (R)
1.7013 cm
Incircle radius (apothem) (r)
1.3764 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. Height is the perpendicular distance from a base edge to the opposite vertex.
  • 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

What is the pentagon height measurement?

With a base edge horizontal, height reaches its opposite vertex perpendicularly. It equals circumradius plus apothem.

References

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

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