# Heptagon
A heptagon is a polygon with seven sides and seven interior angles; it is sometimes called a septagon, though that name mixes the Latin prefix *sept-* with the Greek suffix *-gon* and is not recommended by usage authorities[1](https://mathworld.wolfram.com/Heptagon.html). A heptagon whose vertices are equally spaced around a circle and whose sides are all the same length is the regular heptagon, the seven-sided regular polygon with Schläfli symbol \( \{7\} \)[1](https://mathworld.wolfram.com/Heptagon.html) • [2](https://mathworld.wolfram.com/RegularHeptagon.html). Heptagons are classified as regular or irregular and as convex or concave; a regular heptagon is always convex, while a concave heptagon is necessarily irregular[3](https://www.math.net/heptagon).
| Key fact | Detail |
|---|---|
| Sides and angles | 7 sides; interior angles sum to 900°, exterior angles to 360°[3](https://www.math.net/heptagon) |
| Regular heptagon angles | Each interior angle ≈ 128.57°, each exterior angle ≈ 51.43°[3](https://www.math.net/heptagon) |
| Diagonals and triangles | 14 diagonals; diagonals from one vertex divide it into 5 triangles[3](https://www.math.net/heptagon) |
| Symmetry | 7 lines of symmetry and rotational symmetry of order 7[3](https://www.math.net/heptagon) |
| Area | \( A = \tfrac{7}{4} s^{2} \cot(\pi/7) \approx 3.633\,s^{2} \) for side length \( s \)[4](https://www.statgarden.com/geometry/heptagon.html) |
| Constructibility | The regular heptagon is the smallest regular polygon that cannot be constructed with compass and straightedge[5](https://proofwiki.org/wiki/Regular_Heptagon_is_Smallest_with_no_Compass_and_Straightedge_Construction) |
| Real-world use | The British 50p and 20p coins are curved heptagons of constant width[6](https://www.royalmint.com/collect/decimalisation/the-science-behind-the-shape-of-the-50p/) |
## Angles, diagonals, and area
The interior angles of any heptagon sum to 900°, which follows from dividing the polygon into five triangles from a single vertex, each contributing 180°[3](https://www.math.net/heptagon). In a regular heptagon each interior angle measures \( \tfrac{(7-2)\cdot 180^{\circ}}{7} \approx 128.57^{\circ} \), and each exterior angle measures \( \tfrac{360^{\circ}}{7} \approx 51.43^{\circ} \)[3](https://www.math.net/heptagon) • [7](https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference). A heptagon has 14 diagonals, four from each vertex, and a regular heptagon has 7 lines of symmetry and rotational symmetry of order 7[3](https://www.math.net/heptagon).
The area of a regular heptagon with side length \( s \) is
\[ A = \frac{7}{4}\, s^{2} \cot\!\left(\frac{\pi}{7}\right) \approx 3.633\, s^{2}, \]
obtained by splitting the polygon into seven identical triangles meeting at the center[4](https://www.statgarden.com/geometry/heptagon.html). Unlike the square, hexagon, or pentagon, this formula cannot be simplified to whole-number roots and square roots: exact trigonometric values such as \( \cos(\pi/7) \) are not expressible with simple surds[4](https://www.statgarden.com/geometry/heptagon.html).
## Why the regular heptagon cannot be constructed
The regular heptagon is the smallest regular polygon that cannot be constructed using only a compass and straightedge[5](https://proofwiki.org/wiki/Regular_Heptagon_is_Smallest_with_no_Compass_and_Straightedge_Construction). The reason is algebraic. Constructing the polygon is equivalent to constructing the point \( (\cos \tfrac{2\pi}{7}, \sin \tfrac{2\pi}{7}) \) from the points \( (0,0) \) and \( (1,0) \)[8](https://proofwiki.org/wiki/Compass_and_Straightedge_Construction_for_Regular_Heptagon_does_not_exist). The number \( 2\cos(2\pi/7) \) satisfies the irreducible cubic polynomial
\[ x^{3} + x^{2} - 2x - 1 = 0, \]
whose degree 3 is not a power of 2, so the angle \( 2\pi/7 \) is not constructible[9](https://proofwiki.org/wiki/Compass_and_Straightedge_Construction_for_Regular_Heptagon_does_not_exist/Proof_2).
The general criterion came from [Carl Friedrich Gauss](https://www.edgepedia.org/carl-friedrich-gauss), who showed in 1796 that a regular polygon is constructible if the number of its sides is a product of distinct Fermat primes and a power of 2; since 7 is not a Fermat prime, the regular heptagon fails the test[10](https://mathshistory.st-andrews.ac.uk/SH/gauss_sh.pdf) • [11](https://www.girlsangle.org/page/bulletin-archive/GABv08n02E.pdf). Pierre Laurent Wantzel completed the theorem in 1837 by proving the converse, that only such polygons are constructible[12](https://arxiv.org/abs/2606.27169).
## Constructions that do work
Although the classical tools fail, the regular heptagon is constructible by a neusis construction, in which a marked ruler is used to slide a marked point onto an arc while the ruler passes through a fixed point; Johnson gave such a construction in 1975, and Conway and Guy gave another in 1996[2](https://mathworld.wolfram.com/RegularHeptagon.html). Bankoff and Garfunkel's 1973 survey discusses a purported discovery of the neusis construction by [Archimedes](https://www.edgepedia.org/archimedes), citing Heath[2](https://mathworld.wolfram.com/RegularHeptagon.html). Other exact methods include seven identical toothpicks arranged to form 1:3:3 triangles (Finlay 1959), folding and knotting a strip of paper (Madachy 1979), and the conchoid of Nicomedes[2](https://mathworld.wolfram.com/RegularHeptagon.html).
Approximate compass-and-straightedge constructions also exist. Dixon (1991) gives constructions for angles very close to \( 360^{\circ}/7 \approx 51.428571^{\circ} \), including \( 2\sin^{-1}(\sqrt{3}/4) \approx 51.317813^{\circ} \) and \( \tan^{-1}(5/4) \approx 51.340192^{\circ} \)[2](https://mathworld.wolfram.com/RegularHeptagon.html).
## Tilings and packing
Regular heptagons cannot tile the Euclidean plane by themselves; only the equilateral triangle, square, and regular hexagon can[13](https://openstax.org/books/contemporary-mathematics/pages/10-5-tessellations). A regular tiling of heptagons would require a hyperbolic surface with negative curvature, where every point is a saddle point[14](https://physicstoday.aip.org/backscatter/physics-today). In the hyperbolic plane the \( \{7,3\} \) tiling places equal-sized regular heptagons with three meeting at each vertex; its symmetry group is the (2,3,7) triangle group, and Klein's quartic curve, the maximally symmetric three-holed [Riemann surface](https://www.edgepedia.org/riemann-surface), can be tiled by 24 regular heptagons[15](https://math.ucr.edu/home/baez/visual_insight/20140715%20-%20%7B7%2C3%7D%20Tiling.html).
The heptagon also has a distinctive packing property. The densest known packing of the regular heptagon has density
\[ \frac{2}{97}\left(-111 + 492\cos\!\left(\frac{\pi}{7}\right) - 356\cos^{2}\!\left(\frac{\pi}{7}\right)\right) = 0.89269\ldots, \]
which Greg Kuperberg and Włodzimierz Kuperberg proved in 1990 to be the densest double-lattice packing[16](https://blogs.ams.org/visualinsight/2014/11/15/packing-regular-heptagons/). Of all convex shapes, the regular heptagon is believed to have the lowest maximal packing density, though this has not been proved; it is a proven local minimum of the double-lattice packing density[16](https://blogs.ams.org/visualinsight/2014/11/15/packing-regular-heptagons/) • [17](https://ykallus.github.io/gt2014.html). For comparison, the disk packs at density \( \pi/\sqrt{12} \approx 0.9069 \) and the regular octagon at about 0.9062[16](https://blogs.ams.org/visualinsight/2014/11/15/packing-regular-heptagons/).
## Heptagons in coins and elsewhere
The most familiar heptagons are British coins. The 50 pence and 20 pence pieces are equilateral curve heptagons: seven-sided shapes whose sides are curved arcs rather than straight lines[18](https://www.mathopenref.com/heptagon.html). Hugh Conway, the only technical member of the Decimal Currency Board and Managing Director of Bristol Siddeley Engines, proposed a multisided coin that could roll smoothly without changing height, an essential requirement given the spread of vending machines and parking meters[6](https://www.royalmint.com/collect/decimalisation/the-science-behind-the-shape-of-the-50p/). Early trial pieces included ten- and twelve-sided versions and a rounded square, but the seven-sided design was preferred because it was easily distinguishable from other coins by feel and by sight, and as a coin of constant width it rolls smoothly[6](https://www.royalmint.com/collect/decimalisation/the-science-behind-the-shape-of-the-50p/). Introduced in 1969, it was the world's first seven-sided coin, and [Gibraltar](https://www.edgepedia.org/gibraltar), Tonga, the [Isle of Man](https://www.edgepedia.org/isle-of-man), and the [Falkland Islands](https://www.edgepedia.org/falkland-islands) have since issued coins of the same shape[19](https://www.royalmint.com/stories/collect/the-story-of-the-50p-coin/) • [6](https://www.royalmint.com/collect/decimalisation/the-science-behind-the-shape-of-the-50p/).
The heptagonal triangle, the unique scalene triangle formed from three vertices of a regular heptagon, has vertex angles \( \pi/7 \), \( 2\pi/7 \), and \( 4\pi/7 \), and was the subject of a 1973 survey by Bankoff and Garfunkel[20](https://mathworld.wolfram.com/HeptagonalTriangle.html). In English, Robert Recorde used the word "septangle" in 1551, and "heptagon" appears in 1570 in Sir Henry Billingsley's translation of Euclid's *Elements*[21](https://mathshistory.st-andrews.ac.uk/Miller/mathword/h/).
## References
1. [Weisstein, Eric W. "Heptagon." MathWorld](https://mathworld.wolfram.com/Heptagon.html)
2. [Weisstein, Eric W. "Regular Heptagon." MathWorld](https://mathworld.wolfram.com/RegularHeptagon.html)
3. [Heptagon, Math.net](https://www.math.net/heptagon)
4. [Heptagon Calculator, StatGarden](https://www.statgarden.com/geometry/heptagon.html)
5. [Regular Heptagon is Smallest with no Compass and Straightedge Construction, ProofWiki](https://proofwiki.org/wiki/Regular_Heptagon_is_Smallest_with_no_Compass_and_Straightedge_Construction)
6. [The Science Behind the Shape of the 50p, The Royal Mint](https://www.royalmint.com/collect/decimalisation/the-science-behind-the-shape-of-the-50p/)
7. [10.4 Polygons, Perimeter, and Circumference, OpenStax Contemporary Mathematics](https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference)
8. [Compass and Straightedge Construction for Regular Heptagon does not exist, ProofWiki](https://proofwiki.org/wiki/Compass_and_Straightedge_Construction_for_Regular_Heptagon_does_not_exist)
9. [Compass and Straightedge Construction for Regular Heptagon does not exist/Proof 2, ProofWiki](https://proofwiki.org/wiki/Compass_and_Straightedge_Construction_for_Regular_Heptagon_does_not_exist/Proof_2)
10. [Gauss, sci_hi, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/SH/gauss_sh.pdf)
11. [december 2014 • volume 8 • number 2](https://www.girlsangle.org/page/bulletin-archive/GABv08n02E.pdf)
12. [Theorem of Wantzel, arXiv](https://arxiv.org/abs/2606.27169)
13. [10.5 Tessellations, OpenStax Contemporary Mathematics](https://openstax.org/books/contemporary-mathematics/pages/10-5-tessellations)
14. [Back Scatter, Physics Today (August 2019)](https://physicstoday.aip.org/backscatter/physics-today)
15. [{7,3} Tiling, Visual Insight, AMS](https://math.ucr.edu/home/baez/visual_insight/20140715%20-%20%7B7%2C3%7D%20Tiling.html)
16. [Packing Regular Heptagons, Visual Insight, AMS](https://blogs.ams.org/visualinsight/2014/11/15/packing-regular-heptagons/)
17. [Pessimal Packing Shapes, Yoav Kallus](https://ykallus.github.io/gt2014.html)
18. [Heptagon (7-gon), Math Open Reference](https://www.mathopenref.com/heptagon.html)
19. [The Story of the 50p Coin, The Royal Mint](https://www.royalmint.com/stories/collect/the-story-of-the-50p-coin/)
20. [Weisstein, Eric W. "Heptagonal Triangle." MathWorld](https://mathworld.wolfram.com/HeptagonalTriangle.html)
21. [Earliest Known Uses of Some of the Words of Mathematics (H), MacTutor](https://mathshistory.st-andrews.ac.uk/Miller/mathword/h/)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Elementary and Euclidean geometry › Plane figures*
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