POLYGON
, in Geometry, a figure of many angles; and consequently of many sides also; for every figure has as many sides as angles. If the angles be all equal among themselves, the polygon is said to be a regular one; otherwise, it is irregular. Polygons also take particular names according to the number of their sides; thus a Polygon of 3 sides is called a trigon, 4 sides " a tetragon, 5 sides " a pentagon, 6 sides " a hexagon, &c. and a circle may be considered as a Polygon of an infinite number of small sides, or as the limit of the Polygons.
Polygons have various properties, as below:
1. Every Polygon may be divided into as many triangles as it hath sides.
2. The angles of any Polygon taken together, make twice as many right angles, wanting 4, as the figure hath sides. Thus, if the Polygon has 5 sides; the double of that is 10, from which subtracting 4, leaves 6 right angles, or 540 degrees, which is the sum of the 5 angles of the pentagon. And this property, as well as the former, belongs to both regular and irregular Polygons.
3. Every regular Polygon may be either inscribed in a circle, or described about it. But not so of the irregular ones, except the triangle, and another particular case as in the following property.
An equilateral figure inscribed in a circle, is always equiangular.—But an equiangular figure inscribed in a circle is not always equilateral, but only when the number of sides is odd. For if the sides be of an even number, then they may either be all equal; or else half of them may be equal, and the other half equal to each other, but different from the former half, the equals being placed alternately.
4. Every Polygon, circumscribed about a circle, is equal to a right-angled triangle, of which one leg is the radius of the circle, and the other the perimeter or sum of all the sides of the Polygon. Or the Polygon is equal to half the rectangle under its perimeter and the radius of its inscribed circle, or the perpendicular from its centre upon one side of the Polygon.
Hence, the area of a circle being less than that of its circumscribing Polygon, and greater than that of its inscribed Polygon, the circle is the limit of the inscribed and circumscribed Polygons: in like manner the circumference of the circle is the limit between the perimeters of the said Polygons: consequently the circle is equal to a right-angled triangle, having one leg equal to the radius, and the other leg equal to the circumference; and therefore its area is found by multiplying half the circumference by half the diameter. In like manner, the area of any Polygon is found by multiplying half its perimeter by the perpendicular demitted from the centre upon one side.
5. The following Table exhibits the most remarkable particulars in all the Polygons, up to the dodecagon of 12 sides; viz, the angle at the centre AOB, the angle of the Polygon C or CAB or double of OAB, and the area of the Polygon when each side AB is 1. (See the following figure.)
| No. of sides. | Name of Polygon. | Ang. O at cent. | Ang. C. of Polyg. | Area. |
| 3 | Trigon | 120° | 60° | 0.4330127 |
| 4 | Tetragon | 90 | 90 | 1.0000000 |
| 5 | Pentagon | 72 | 108 | 1.7204774 |
| 6 | Hexagon | 60 | 120 | 2.5980762 |
| 7 | Heptagon | 51 3/7 | 128 4/7 | 3.6339124 |
| 8 | Octagon | 45 | 135 | 4.8284271 |
| 9 | Nonagon | 40 | 140 | 6.1818242 |
| 10 | Decagon | 36 | 144 | 7.6942088 |
| 11 | Undecagon | 32 8/11 | 147 3/11 | 9.3656399 |
| 12 | Dodecagon | 30 | 150 | 11.1961524 |
By means of the numbers in this Table, any Polygons may be constructed, or their areas found: thus, (1st) To inscribe a Polygon in a given Circle. At the centre make the angle O equal to the angle at the centre of the proposed Polygon, found in the 3d colum<*> of the Table, the legs cutting the circle in A and B; and join A and B which will be one side of the Polygon. Then take AB between the compasses, and apply it continually round the circumference, to complete the Polygon.
(2d) Upon the given Line AB to describe a regular Polygon. From the extremities draw the two lines AO and BO, making the angles A and B each equal to half the angle of the Polygon, found in the 4th column of the Table, and their intersection O will be the centre of the circumscribed circle: then apply AB continually round the circumference as before.
(3d) To describe a Polygon about a given Circle.— At the centre O make the angle of the centre as in the 1st art. its legs cutting the circle in a and b: join ab, and parallel to it draw AB to touch the circle: and meeting Oa and Ob produced in A and B: with the radius OA, or OB, describe a circle, and around its circumference apply continual AB, which will complete the P<*>lygon as before.
(4th) To find the Area of any regular Polygon.— Multiply the square of its side by the tabular area, found on the line of its name in the last column of the Table, and the product will be the area. Thus, to|
| 0.4330127 |
| 400 |
| 173.2050800 |
6. There are several curious algebraical theorems for inscribing Polygons in circles, or finding the chord of any proposed part of the circumference, which is the same as angular sections. These kinds of sections, or parts and multiples of arcs, were sirst treated of by Vieta, as shewn in the Introduction to my Log. pa. 9, and since pursued by several other mathematicians, in whose works they are usually to be found. Many other particulars relating to Polygons may also be seen in my Mensuration, 2d edit. pa. 20, 21, 22, 23, 113, &c.
Polygon, in Fortisication, denotes the figure or perimeter of a fortress, or fortified place. This is either Exterior or Interior.
Exterior Polygon is the perimeter or figure formed by lines connecting the points of the bastions to one another, quite round the work. And
Interior Polygon, is the perimeter or figure formed by lines connecting the centres of the bastions, quite around.
Line of Polygons, is a line on some sectors, containing the homologous sides of the first nine regular Polygons inscribed in the same circle; viz, from an equilateral triangle to a dodecagon.
POLYGONAL Numbers, are the continual or successive sums of a rank of any arithmeticals beginning at 1, and regularly increafing; and therefore are the first order of figurate numbers; they are called Polygonals, because the number of points in them may be arranged in the form of the several Polygonal figures in geometry, as is illustrated under the article Figurate Numbers, which see.
The several sorts of Polygonal numbers, viz, the triangles, squares, pentagons, hexagons, &c, are formed from the addition of the terms of the arithmetical series, having respectively their common difference 1, 2, 3, 4, &c; viz, if the common difference of the arithmeticals be 1, the sums of their terms will form the triangles; if 2, the squares; if 3, the pentagons; if 4, the hexagons, &c. Thus:
| { | Arith. Progres. | 1 , | 2 , | 3 , | 4 , | 5 , | 6 , | 7 . |
| Triang. Nos. | 1 , | 3 , | 6 , | 10 , | 15 , | 21 , | 28 . | |
| { | Arith. Progres. | 1 , | 3 , | 5 , | 7 , | 9 , | 11 , | 13 . |
| Square Numbers | 1 , | 4 , | 9 , | 16 , | 25 , | 36 , | 49 . | |
| { | Arith. Progres. | 1 , | 4 , | 7 , | 10 , | 13 , | 16 , | 19 . |
| Pentagonal Nos. | 1 , | 5 , | 12 , | 22 , | 35 , | 51 , | 70 . | |
| { | Arith. Progres. | 1 , | 5 , | 9 , | 13 , | 17 , | 21 , | 25 . |
| Hexagonal Nos. | 1 , | 6 , | 15 , | 28 , | 45 , | 66 , | 91 . |
The Side of a Polygonal number is the number of points in each side of the Polygonal figure when the points in the number are ranged in that form. And this is also the same as the number of terms of the arithmeticals that are added together in composing the Po- lygonal number; or, in short, it is the number of the term from the beginning. So, in the 2d or squares, 1 2 3 4 1 4 9 16 the side of the first (1) is 1, that of the second (4) is 2, that of the third (9) is 3, that of the fourth (16) is 4, and so on. And
The Angles, or Numbers of Angles, are the same as those of the figure from which the number takes its name. So the angles of the triangular numbers are 3, of the square ones 4, of the pentagonals 5, of the hexagonals 6, and so on. Hence, the angles are 2 more than the common difference of the arithmetical series from which any rank of Polygonals is formed: so the arithmetical series has for its common difference the number 1 or 2 or 3 &c as follows, viz, 1 in the triangles, 2 in the squares, 3 in the pentagons, &c; and, in general, if a be the number of angles in the Polygon, then a - 2 is = d the common difference of the arithmetical series, or the number of angles.
Prob. 1. To find any Polygonal Number proposed; having given its side n and angles a. The Polygonal number being evidently the sum of the arithmetical progression whose number of terms is n and common difference a - 2, and the sum of an arithmetical progression being equal to half the product of the extremes by the number of terms, the extremes being 1 and ; therefore that number, or this sum, will be or , where d is the common difference of the arithmeticals that form the Polygonal number, and is always 2 less than the number of angles a.
Hence, for the several sorts of Polygons, any particular number whose side is n, will be found from either of these two formulæ, by using for d its values 1, 2, 3, 4, &c; which gives these following formulæ for the Polygonal number in each sort, viz, the
| Triangular | , |
| Square | , |
| Pentagonal | , |
| Hexagonal | , |
| Heptagonal | , |
| &c. |
Prob. 2. To find the Sum of any Number of Polygonal Numbers of any order.—Let the angles of the Polygon| be a, or the common difference of the arithmeticals that form the Polygonals, d; and n the number of terms in the Polygonal series, whose sum is sought: then is or the sum of the n terms sought.
Hence, substituting successively the numbers 1, 2, 3, 4, &c, for d, there is obtained the following particular cases, or formulæ, for the sums of n terms of the several ranks of Polygonal numbers, viz, the sum of the
| Triangulars | , |
| Squares | , |
| Pentagonals | , |
| Hexagonals | , |
| Heptagonals | , |
| &c |
