Calculate area, perimeter, apothem, diagonals, circumscribed and inscribed circles for a regular heptagon. Enter any one measurement to find all properties.
The heptagon — a seven-sided polygon — appears less frequently in everyday life than triangles, squares, or hexagons, yet it holds a fascinating place in geometry and design. Regular heptagons feature prominently in coin design: the British 50-pence and 20-pence coins are Reuleaux heptagons (constant-width curves based on heptagonal geometry), chosen because vending machines can identify them regardless of orientation.
A regular heptagon has all seven sides equal and all interior angles equal to approximately 128.571° — a number that cannot be expressed as a simple fraction of 360°. This makes the heptagon one of the polygons that cannot be constructed with compass and straightedge alone (proven by Carl Friedrich Gauss). Each exterior angle measures roughly 51.429°, and the sum of interior angles is exactly 900°.
This calculator computes every property of a regular heptagon from a single known measurement: side length, area, perimeter, apothem, or circumradius. It reveals all 14 diagonals (split into short and long types), the circumscribed and inscribed circles, and provides real-world references. Whether you are designing a gazebo, studying polygon geometry, or exploring coin mathematics, this tool has you covered.
Calculating heptagon properties by hand involves non-trivial trigonometric functions — cot(π/7) and sin(π/7) produce irrational numbers that don't simplify neatly. This calculator eliminates the complexity: enter any single measurement and instantly receive all dimensions, angles, diagonal lengths, and circle properties.
Whether you're an architect designing a seven-sided pavilion, a student exploring polygon geometry, a numismatist studying heptagonal coins, or a craftsperson laying out a seven-sided frame, this tool provides every number you need with clear formulas and visual comparisons.
Area = (7/4)s² cot(π/7). Perimeter = 7s. Apothem a = s / (2 tan(π/7)). Circumradius R = s / (2 sin(π/7)). Interior angle = (n−2)×180°/n ≈ 128.571°. Number of diagonals = n(n−3)/2 = 14. Where s = side length and n = 7.
Result: Area ≈ 36.34 cm², Perimeter = 35 cm, Apothem ≈ 5.19 cm, Circumradius ≈ 5.78 cm
For a regular heptagon with side 5 cm: Area = (7/4)(25)cot(π/7) ≈ 36.34 cm². Perimeter = 7 × 5 = 35 cm. Apothem = 5/(2 tan(π/7)) ≈ 5.19 cm. Circumradius = 5/(2 sin(π/7)) ≈ 5.78 cm. It has 14 diagonals, short diag ≈ 8.68 cm and long diag ≈ 10.38 cm.
The regular heptagon occupies a special niche in polygon geometry. While triangles (3), squares (4), and hexagons (6) can tile the plane, and pentagons (5) appear in nature through the golden ratio, the heptagon stands apart: it cannot be constructed with compass and straightedge, it cannot tile the plane, and its angles are irrational multiples of π degrees.
Despite these limitations, the heptagon appears in coin design, architecture, and recreational mathematics. Its interior angle of about 128.57° is close to the 120° of a hexagon but different enough to create distinct visual patterns.
A regular heptagon has 14 diagonals, which fall into two distinct length classes. The "short" diagonals connect vertices separated by one vertex, with length d₁ = 2R sin(2π/7). The "long" diagonals connect vertices separated by two vertices, with length d₂ = 2R sin(3π/7). The ratio d₂/d₁ is a constant approximately equal to 1.247.
Interestingly, these diagonal lengths satisfy specific polynomial relationships that connect to the minimal polynomial of cos(2π/7), which is 8x³ + 4x² − 4x − 1 = 0.
Although perfect heptagons are rare in nature, approximate seven-fold symmetry appears in some flower species and mineral formations. In human-designed systems, heptagonal geometry is used for security features in currency (the Reuleaux heptagon rolls smoothly in vending machines), decorative rosettes, and architectural follies. The seven-sided design offers a pleasing balance between the common hexagon and octagon, providing variety in tiling patterns when combined with other shapes.
A regular heptagon is a polygon with seven equal sides and seven equal interior angles, each measuring approximately 128.571°.
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.
No. Gauss proved that a regular polygon is constructible only if its number of sides is a product of a power of 2 and distinct Fermat primes. 7 is not a Fermat prime, so the heptagon cannot be constructed exactly with compass and straightedge.
The sum of interior angles is (7−2) × 180° = 900°. Keep this note short and outcome-focused for reuse.
The UK 50p and 20p coins are Reuleaux heptagons. Some garden gazebos, decorative tiles, and architectural elements also use seven-sided designs.
The apothem is the perpendicular distance from the center to the midpoint of a side (also the inscribed circle radius). The circumradius is the distance from the center to a vertex (the circumscribed circle radius).
A regular heptagon can be divided into 7 isosceles triangles from the center. Each triangle has area (1/2)·s·a, where a is the apothem. Total area = (7/2)·s·a = (7/4)s² cot(π/7).