Heptagon

In geometry, a heptagon 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​4⁄7 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 π / 7 , {displaystyle pi /7,} and the area of each of the 14 small triangles is one-fourth of the apothem. The exact algebraic expression, starting from the cubic polynomial x3 + x2 − 2x − 1 (one of whose roots is 2 cos ⁡ 2 π 7 {displaystyle 2cos { frac {2pi }{7}}} ) is given in complex numbers by: in which the imaginary parts offset each other leaving a real-valued expression. This expression cannot be algebraically rewritten without complex components, since the indicated cubic function is casus irreducibilis. The area of a regular heptagon inscribed in a circle of radius R is 7 R 2 2 sin ⁡ 2 π 7 , {displaystyle { frac {7R^{2}}{2}}sin { frac {2pi }{7}},} while the area of the circle itself is π R 2 ; {displaystyle pi R^{2};} thus the regular heptagon fills approximately 0.8710 of its circumscribed circle.