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Concept
Heptagon
Summary
In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon.
The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of septua-, a Latin-derived numerical prefix, rather than hepta-, a Greek-derived numerical prefix; both are cognate) together with the Greek suffix "-agon" meaning angle.
A regular heptagon, in which all sides and all angles are equal, has internal angles of 5π/7 radians (128 degrees). Its Schläfli symbol is {7}.
The area (A) of a regular heptagon of side length a is given by:
This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with vertices at the center and at the heptagon's vertices, and then halving each triangle using the apothem as the common side. The apothem is half the cotangent of and the area of each of the 14 small triangles is one-fourth of the apothem.
The area of a regular heptagon inscribed in a circle of radius R is while the area of the circle itself is thus the regular heptagon fills approximately 0.8710 of its circumscribed circle.
As 7 is a Pierpont prime but not a Fermat prime, the regular heptagon is not constructible with compass and straightedge but is constructible with a marked ruler and compass. It is the smallest regular polygon with this property. This type of construction is called a neusis construction. It is also constructible with compass, straightedge and angle trisector. The impossibility of straightedge and compass construction follows from the observation that is a zero of the irreducible cubic x3 + x2 − 2x − 1. Consequently, this polynomial is the minimal polynomial of whereas the degree of the minimal polynomial for a constructible number must be a power of 2.
An approximation for practical use with an error of about 0.2% is to use half the side of an equilateral triangle inscribed in the same circle as the length of the side of a regular heptagon. It is unknown who first found this approximation, but it was mentioned by Heron of Alexandria's Metrica in the 1st century AD, was well known to medieval Islamic mathematicians, and can be found in the work of Albrecht Dürer.
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