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Heptagon Area from Linear Measurements

Work between a regular heptagon’s area, side, perimeter, outer radius and apothem using its fixed seven-sided geometry.

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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.

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Formula and method

A regular 7-sided outline with side 2 cm has perimeter 14 cm. Its area follows by dividing it into 7 congruent triangles from the center, each with base 2 cm and height equal to the apothem.

P = na; r = a / [2 tan(π/n)]; R = a / [2 sin(π/n)]; A = Pr/2; α = (n−2)π/n; β = 2π/n

Worked example

Enter these known values and leave the other values blank.

Equal side length a
2 cm
Interior angle α
128.57 deg
Central / exterior angle β
51.43 deg
Perimeter around the outline
14 cm
Outer circle radius R
2.3048 cm
Apothem / inner circle radius r
2.0765 cm
Enclosed area
14.536 cm²

Assumptions and limitations

  • The outline must be a convex regular polygon: every side has equal length and every interior angle is equal. An irregular outline needs additional measurements.
  • Use a shared length unit for linear measurements. Area uses the square of a length unit, so check its separate unit selector.
  • The apothem is the perpendicular center-to-edge distance; the circumradius reaches a vertex. They are different quantities.
  • The central angle β equals the exterior turning angle, while the interior angle is α. Their sum is 180°.
  • Reverse calculations require enough independent measurements. Keep the side count locked when the intended shape must remain fixed.
  • The named side count is fixed in this model. Zero-size outputs, where accepted by the captured conditions, represent a degenerate limit rather than a positive-area shape.

Common questions

Why does area equal half the perimeter times the apothem?

Segments from the center to each vertex partition the polygon into congruent triangles. Adding their base times height divided by two gives the perimeter times the apothem divided by two.

What changes when all lengths double?

Perimeter, radii and diagonals double, while area becomes four times as large. Interior and central angles stay the same.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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