Heptagon Calculator
A regular heptagon is a seven-sided polygon with all sides and all interior angles equal. Choose a known measurement, enter its value, and get the area, perimeter, apothem (inradius), circumradius, both diagonal lengths, and interior angle instantly.
Formula
Worked example
For side a = 10 cm: P = 70 cm; apothem r = 10 / (2·tan(π/7)) ≈ 10.3826 cm; circumradius R = 10 / (2·sin(π/7)) ≈ 11.5238 cm; area A = (7/4)·100·cot(π/7) ≈ 363.39 cm². Short diagonal d2 = 2·11.5238·sin(2π/7) ≈ 18.019 cm; long diagonal d3 = 2·11.5238·sin(3π/7) ≈ 22.252 cm. Each interior angle is 900°/7 ≈ 128.571°.
How the heptagon area formula works
A regular heptagon can be divided from its centre into seven identical isosceles triangles. Each triangle has a base equal to one side a and a height equal to the apothem r, the perpendicular distance from the centre to the midpoint of that side. Because r = (a/2)·cot(π/7), each triangle has area (1/2)·a·r = (a²/4)·cot(π/7), and seven of them give the total A = (7/4)·a²·cot(π/7). The cotangent of π/7 (25.714°) is approximately 2.0765, making the constant factor about 3.6339. Because the side is squared, doubling the side length multiplies the area by four.
Angles: interior, exterior, and the sum rule
For any simple polygon with n sides, the interior angles sum to (n - 2) × 180°. For a heptagon, that is 5 × 180° = 900°. In a regular heptagon all seven angles are equal, so each measures 900°/7 ≈ 128.571°. The exterior angle at each vertex is 360°/7 ≈ 51.429°, and the seven exterior angles always sum to exactly 360°. The heptagon is notable for being the smallest regular polygon that cannot be constructed exactly with compass and straightedge alone: because 7 is not a Fermat prime, no exact Euclidean construction exists.
Diagonals: short, long, and the count
A regular heptagon has n(n - 3)/2 = 7 × 4/2 = 14 diagonals. They come in two distinct lengths. The short diagonal connects a vertex to the vertex two steps away around the polygon: d2 = 2R·sin(2π/7). The long diagonal connects a vertex to the vertex three steps away (the furthest possible in a 7-gon, since the vertex four steps away is the same as three steps in the other direction): d3 = 2R·sin(3π/7). For a unit side, d2 ≈ 1.802 and d3 ≈ 2.247.
Reverse-solve: starting from the apothem or circumradius
If you know the apothem (inscribed circle radius r), the side length is a = 2r·tan(π/7) ≈ 0.9634·r×2. If you know the circumradius (circumscribed circle radius R), the side length is a = 2R·sin(π/7) ≈ 0.8678·R. The calculator performs these conversions automatically when you switch the "Solve from" selector, then derives every other dimension from the recovered side length. This is useful for fitting a heptagon inside a known circle or around a known inscribed circle.
Regular heptagon dimensions by side length
| Side (a) | Perimeter | Apothem (r) | Circumradius (R) | Short diag (d2) | Long diag (d3) | Area |
|---|---|---|---|---|---|---|
| 1 | 7 | 1.0383 | 1.1524 | 1.8019 | 2.247 | 3.6339 |
| 5 | 35 | 5.1913 | 5.7617 | 9.0094 | 11.2353 | 90.848 |
| 10 | 70 | 10.3826 | 11.5238 | 18.019 | 22.47 | 363.39 |
| 15 | 105 | 15.5739 | 17.2857 | 27.029 | 33.706 | 817.62 |
| 20 | 140 | 20.7652 | 23.0476 | 36.038 | 44.941 | 1453.6 |
| 50 | 350 | 51.913 | 57.617 | 90.094 | 112.35 | 9084.8 |
All values computed from A = (7/4)a²·cot(π/7). Diagonals use d2 = 2R·sin(2π/7) and d3 = 2R·sin(3π/7).
Frequently asked questions
What is the area formula for a regular heptagon?
For a regular heptagon with side length a, the area is A = (7/4)·a²·cot(π/7) ≈ 3.6339·a². You can also write it as A = (7/2)·R²·sin(2π/7) in terms of the circumradius R, or A = (7/2)·r²·tan(π/7) in terms of the apothem r.
How many degrees are in each interior angle of a heptagon?
The interior angles of any heptagon sum to (7 - 2) × 180° = 900°. In a regular heptagon each angle is 900°/7 ≈ 128.571°. Each exterior angle is 360°/7 ≈ 51.429°.
What is the difference between the apothem and the circumradius?
The apothem (inradius r) is the perpendicular distance from the centre to the midpoint of a side: r = a/(2·tan(π/7)). It is the radius of the inscribed circle that touches all seven sides. The circumradius R = a/(2·sin(π/7)) is the distance from the centre to a vertex, and is the radius of the circumscribed circle that passes through all seven corners. The circumradius is always larger than the apothem.
How many diagonals does a heptagon have, and what are their lengths?
A heptagon has n(n - 3)/2 = 14 diagonals. They come in two lengths: 7 short diagonals connecting vertices two apart (d2 = 2R·sin(2π/7) ≈ 1.802·a) and 7 long diagonals connecting vertices three apart (d3 = 2R·sin(3π/7) ≈ 2.247·a).
Can I calculate a heptagon from its circumradius or apothem instead of the side?
Yes. Switch the "Solve from" selector to "Apothem" or "Circumradius", enter the known value, and the calculator recovers the side length then derives every other dimension. From the apothem r: a = 2r·tan(π/7). From the circumradius R: a = 2R·sin(π/7).
Can a regular heptagon be constructed with a compass and straightedge?
No. A regular heptagon cannot be constructed exactly with compass and straightedge alone. The reason is that 7 is not a Fermat prime (only 3, 5, 17, 257, and 65537 are). In practice, the heptagon is drawn by measuring the circumradius and marking vertices at exactly 360°/7 ≈ 51.429° intervals using a protractor.