# HEPTAGON

, in Geometry, a figure of seven sides and seven angles.—When those sides and angles are all equal, the Heptagon is said to be regular, otherwise it is irregular.

In a regular Heptagon, the
angle Cat the centre is = 51° 3/7,
the angle DAB of the polygon
is = 128° 4/7, and its half
CAB = 64° 2/7. Also the area
is = the square of the side
or , where *t* is the tangent
of the angle CAB of
64° 2/7 to the radius 1; or *t* is
the root of the equation;
; or
,|
where the value of *x* and *y* are the roots of the equations
.

See my Mensuration, p. 21, 114, and 116, 2d edition.

Heptagon, in Fortification, a place fortified or strengthened with seven bastions for its defence.

Heptagonal *Numbers,* are a kind of polygonal
numbers in which the difference of the terms of the corresponding
arithmetical progression is 5. Thus,

Arithmeticals, | 1, | 6, | 11, | 16, | 21, | 26, | &c. |

Heptagonals, | 1, | 7, | 18, | 34, | 55, | 81, | &c. |

One property, among many others, of these Heptagonal
numbers is, that if any one of them be multiplied by
40, and to the product add 9, the sum will be a square
number.
Thus ;
and ;
and ;
and ; &c.
Where it is remarkable that the series of squares so
formed is 7^{2}, 17^{2}, 27^{2}, 37^{2}, &c, the common difference
of whose roots is 10, the double of the common
difference of the arithmetical series from which the
Heptagonals are formed.—See Polygonals.

HEPTANGULAR *Figure,* in Geometry, is one
that has seven angles; and therefore also seven sides.