Heptagon Area & Properties Calculator

Calculate the area, perimeter, apothem, circumradius, inradius, diagonals, and angles of a regular heptagon (7-sided polygon) from side length, area, perimeter, apothem, or circumradius.

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Planning notes, formulas, and examples

About the Heptagon Area & Properties Calculator

A regular heptagon (also called a septagon) is a polygon with seven equal sides and seven equal interior angles. Each interior angle measures approximately 128.57°, each exterior angle approximately 51.43°, and the polygon has exactly 14 diagonals. The heptagon is one of the more fascinating regular polygons because it is the smallest polygon that cannot be constructed with a compass and straightedge alone — its construction requires a neusis or angle-trisection technique.

The area of a regular heptagon with side length s is given by A = (7/4) × s² × cot(π/7), which is approximately 3.634 × s². This formula follows from dividing the heptagon into seven congruent isosceles triangles, each with a base equal to the side and height equal to the apothem. The apothem — the perpendicular distance from the center to the midpoint of any side — equals (s/2) × cot(π/7). The circumradius (center to vertex) equals s / (2 × sin(π/7)).

Heptagons appear in nature and design: the seven-sided 20-pence and 50-pence coins of the United Kingdom use a curved heptagonal shape (a Reuleaux polygon) for vending machine compatibility. Several national coats of arms and architectural patterns feature heptagonal symmetry. In tiling theory, regular heptagons cannot tile the Euclidean plane but do appear in hyperbolic tessellations.

This calculator lets you solve from five different inputs — side length, area, perimeter, apothem, or circumradius — and computes all key properties including both diagonal lengths, all angles, and comparative data against other regular polygons from triangle through dodecagon.

When This Page Helps

The Heptagon Area & Properties Calculator is useful when you need fast and consistent geometry results without reworking the same algebra repeatedly. It helps you move from raw measurements to Area, Perimeter, Apothem in one pass, with conversions and derived values shown together.

How to Use the Inputs

  1. Select what measurement you know: side length, area, perimeter, apothem, or circumradius.
  2. Choose a measurement unit (mm, cm, in, m, or ft).
  3. Enter the known value in the input field.
  4. Or click a preset button to load a common example.
  5. View area, perimeter, apothem, circumradius, diagonals, and all angles.
  6. Compare the heptagon to other regular polygons in the reference table.
  7. Use the bar chart to visualize relative dimensions.
Formula used
Area: A = (7/4) × s² × cot(π/7) ≈ 3.634 × s² Perimeter: P = 7s Apothem: a = (s/2) × cot(π/7) Circumradius: R = s / (2 sin(π/7)) Inradius: r = apothem Interior angle: (n−2)×180°/n = 128.571° Exterior angle: 360°/n = 51.429° Diagonals: n(n−3)/2 = 14

Example Calculation

Result: Area ≈ 363.39 cm², Perimeter = 70 cm, Apothem ≈ 10.38 cm, Circumradius ≈ 11.52 cm

For a regular heptagon with side 10 cm: Area = (7/4) × 100 × cot(π/7) ≈ 363.39 cm². Perimeter = 7 × 10 = 70 cm. Apothem = 5 × cot(π/7) ≈ 10.38 cm. Circumradius = 10 / (2 sin(π/7)) ≈ 11.52 cm. Short diagonal ≈ 17.98 cm, long diagonal ≈ 21.83 cm. 14 diagonals total.

Tips & Best Practices

  • A regular heptagon is the simplest polygon that cannot be constructed with compass and straightedge alone.
  • UK 20p and 50p coins are curved heptagons — a Reuleaux polygon based on the heptagon.
  • The area coefficient (≈ 3.634 × s²) falls between the hexagon (≈ 2.598 × s²) and the octagon (≈ 4.828 × s²).
  • All 14 diagonals pass through the interior; they come in three different lengths for a regular heptagon.
  • As the number of polygon sides increases, the shape approaches a circle and the area approaches π × R².

How This Heptagon Area & Properties Calculator Works

Where It Helps In Practice

Heptagon Area & Properties Calculator calculations show up in coursework, drafting, construction layout, packaging, tank sizing, machining, and quality control. Instead of solving each transformation manually, you can test scenarios quickly and verify whether your dimensions remain within tolerance.

Accuracy And Setup Tips

Sources & Methodology

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Frequently Asked Questions

  • A heptagon is a polygon with 7 sides. A regular heptagon has all sides equal and all interior angles equal (≈ 128.57° each).

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