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.
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.
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.
Use it to validate homework steps, check CAD or fabrication dimensions, estimate material requirements, and sanity-check hand calculations before submitting work.
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
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.
This calculator combines the core geometry formula with the input mode selected in the interface, then derives companion values shown in the output cards, comparison bars, and reference tables. Use it to cross-check both direct calculations and reverse-solving scenarios where one measurement is unknown.
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.
Keep units consistent across every input before interpreting area, perimeter, angle, or volume outputs. For best results, measure carefully, round only at the final step, and compare at least one manual calculation with the calculator output when building confidence.
A heptagon is a polygon with 7 sides. A regular heptagon has all sides equal and all interior angles equal (≈ 128.57° each).
A = (7/4) × s² × cot(π/7), which simplifies to approximately 3.634 × s², where s is the side length.
A heptagon has 14 diagonals, calculated by n(n − 3)/2 = 7 × 4 / 2 = 14. Use this as a practical reminder before finalizing the result.
Each interior angle is (7 − 2) × 180° / 7 ≈ 128.57°. The exterior angle is 360°/7 ≈ 51.43°.
No. The regular heptagon is the smallest polygon that cannot be constructed with compass and straightedge. It requires a neusis construction or trisection tool.
The apothem is the distance from the center to the midpoint of a side. For side length s, apothem = (s/2) × cot(π/7) ≈ 1.038 × s.