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Polygon

Polygon

A polygon is a geometric figure in two dimensions with three or more sides. The name comes from two Greek words, poly, meaning "many,"

and gon, meaning "angle." A polygon always has as many angles as it has sides. And in general, polygons are named to indicate the number of sides or angles they contain. Thus, a hexagon has six (hexa- means "six") sides and six angles.

Terminology used in describing polygons

Parts and properties of polygons.

Side: Any one of the straight lines that make up the polygon.

Vertex: A point where any two of the sides of a polygon meet to form an angle.

Angle: A figure formed by the intersection of two sides.

Diagonal: A line that joins any two nonadjacent (not next to each other) vertices.

Perimeter: The sum of the length of all sides.

Area: The space enclosed within the polygon.

Types of polygons.

Equilateral: A polygon in which all sides are equal in length.

Equiangular: A polygon in which all angles are the same size.

Regular: A polygon that is both equilateral and equiangular.

Examples of polygons

The most common kinds of polygons include:

Parallelogram: A quadrilateral (four-sided figure) in which both pairs of sides are parallel and equal.

Rhombus: A parallelogram in which all four sides are equal.

Rectangle: A parallelogram in which all angles are right angles.

Square: A rectangle in which all four sides are equal.

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polygon

polygon, closed plane figure bounded by straight line segments as sides. A polygon is convex if any two points inside the polygon can be connected by a line segment that does not intersect any side. If a side is intersected, the polygon is called concave. In a regular polygon the sides are of equal length and meet at equal angles; all other polygons are not regular, although either their sides or their angles may be equal, as in the cases of the rhombus and the rectangle. The simplest regular polygons are the equilateral triangle, the square, the regular pentagon (of 5 sides), and the regular hexagon (of 6 sides). Although the Greeks had developed methods of constructing these four polygons using only a straightedge and compass, they were unable to do the same for the regular heptagon (of 7 sides). In the 19th cent. C. F. Gauss showed that a regular heptagon was impossible to construct in this way. He proved that a regular polygon is constructible with a straightedge and compass only when the number of sides p is a prime number (see number theory) of the form p = 22n + 1 or a product of such primes. The first five regular polygons with a prime number of sides that can be constructed using a straightedge and compass have 3, 5, 17, 257, and 65,537 sides.

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polygons

polygons

Name

Number of sides

Each internal angle

Sum of internal angles

Triangle

3

60°

180°

Square

4

90°

360°

Pentagon

5

108°

540°

Hexagon

6

120°

720°

Heptagon

7

128.6°

900°

Octagon

8

135°

1080°

Nonagon

9

140°

1260°

Decagon

10

144°

1440°

Undecagon

11

147.3°

1620°

Dodecagon

12

150°

1800°


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polygon

polygon Plane geometric figure having three or more sides intersecting at three or more points (vertices). They are named according to the number of sides or vertices: triangle (three-sided), quadrilateral (four-sided), hexagon (six-sided). A regular polygon is equilateral (has sides equal in length) and equiangular (has equal angles).

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polygon

pol·y·gon / ˈpäliˌgän/ • n. Geom. a plane figure with at least three straight sides and angles, and typically five or more. DERIVATIVES: po·lyg·o·nal / pəˈligənl/ adj.po·lyg·o·nal·ly adv.

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polygon

polygon •deafen •griffon, stiffen •antiphon •hyphen, siphon •often, soften •orphan • ibuprofen •roughen, toughen •colophon •dragon, flagon, lagan, pendragon, wagon •snapdragon • bandwagon • jargon •Megan •Copenhagen, pagan, Reagan •Nijmegen •Antiguan, Egan, Keegan, Regan, vegan •Wigan • cardigan • Milligan • polygon •hooligan • mulligan • ptarmigan •Branigan • Oregon • Michigan •Rattigan •tigon, trigon •toboggan •Glamorgan, gorgon, Morgan, morgen, organ •Brogan, hogan, Logan, slogan •Cadogan • decagon •Aragon, paragon, tarragon •hexagon • pentagon • heptagon •octagon • Bergen • Spitsbergen

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Polygons

Polygons

Classification

Angles

Polygons are closed plane figures bounded by three or more line segments. In the world of geometry, polygons abound. The term refers to a multisided geometric form in the plane. The number of angles in a polygon always equals the number of sides. Polygons are named to indicate the number of their sides or number of noncollinear points present in the polygon (Table 1).

A square is a special type of polygon, as are triangles, parallelograms, and octagons. The prefix of the term, poly comes from the Greek word for many, and the root word gon comes from the Greek word for angle.

Classification

A regular polygon is one whose sides and interior angles are congruent. Regular polygons can be inscribed by a circle such that the circle is tangent to the sides at the centers, and circumscribed by a circle such that the sides form chords of the circle. Regular polygons are named to indicate the number of their sides or number of vertices present in the figure. thus, a hexagon has six sides, while a decagon has ten sides. Examples of regular polygons also include the familiar square and octagon (Figure 1).

Not all polygons are regular or symmetric. Polygons for which all interior angles are less than 180°are called convex. Polygons with one or more interior angles greater than 180° are called concave.

Table 1. Polygons (Thomson Gale. )
Polygons
Name of the polygon Number of sides in polygon Number of vertices of polygon
Triangle33
Rectangle44
Pentagon55
Hexagon66
Heptagon77
Octagon88
Nonagon99
Decagon1010
n-gonnn

The most common image of a polygon is of a multisided perimeter enclosing a single, uninterrupted area. In reality, the sides of a polygon can intersect to form multiple, distinct areas. Such a polygon is classified as reflex.

Angles

In a polygon, the line running between nonadja-cent points is known as a diagonal. The diagonals drawn from a single vertex to the remaining vertices in an n-sided polygon will divide the figure into n2 triangles. The sum of the interior angles of a convex polygon is then just (n2)×180.

If the side of a polygon is extended past the intersecting adjacent side, it defines the exterior angle of the vertex. Each vertex of a convex polygon has two possible exterior angles, defined by the continuation of each of the sides. These two angles are congruent,

KEY TERMS

Angle A geometric figure created by two lines drawn from the same point.

Concave A polygon whose at least one angle is larger than the straight angle (180°).

Convex A polygon whose all angles are less than the straight angle (180°).

Diagonal The line which links-connects any two non-adjacent vertices.

Equiangular A polygon is equiangular if all of its angles are identical.

Equilateral A polygon is equilateral if all the sides are equal in length.

Parallelogram A rectangle with both pair of sides parallel.

Perimeter The sum of the length of all sides.

Rectangle A parallelogram in which all angles are right angles.

Reflex polygon A polygon in which two nonadja-cent sides intersect.

Regular polygon An equilateral, equiangular polygon.

Rhombus A parallelogram whose adjacent sides are equal.

Square A four-sided shape whose sides are equal.

Vertex The point at which the two sides of an angle meet.

however, so the exterior angle of a polygon is defined as one of the two angles. The sum of the exterior angles of any convex polygon is equal to 360°.

Kristin Lewotsky

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Polygons

Polygons

Polygons are closed plane figures bounded by three or more line segments. In the world of geometry , polygons abound. The term refers to a multisided geometric form in the plane. The number of angles in a polygon always equals the number of sides.

Polygons are named to indicate the number of their sides or number of noncollinear points present in the polygon.

A square is a special type of polygon, as are triangles, parallelograms, and octagons. The prefix of the term, poly comes from the Greek word for many, and the root word Gon comes from the Greek word for angle .


Classification

A regular polygon is one whose whose sides and interior angles are congruent. Regular polygons can be inscribed by a circle such that the circle is tangent to the sides at the centers, and circumscribed by a circle such that the sides form chords of the circle. Regular polygons are named to indicate the number of their sides or number of vertices present in the figure. Thus, a hexagon has six sides, while a decagon has ten sides. Examples of regular polygons also include the familiar square and octagon.

Not all polygons are regular or symmetric. Polygons for which all interior angles are less than 180° are called


TABLE 1 . POLYGONS
Name of the polygon Number of sides in polygon Number of vertices of polygon
Triangle 3 3
Rectangle 4 4
Pentagon 5 5
Hexagon 6 6
Heptagon 7 7
Octagon 8 8
Nonagon 9 9
Decagon 10 10
n-gon n n



convex. Polygons with one or more interior angles greater than 180° are called concave.

The most common image of a polygon is of a multisided perimeter enclosing a single, uninterrupted area. In reality, the sides of a polygon can intersect to form multiple, distinct areas. Such a polygon is classified as reflex .

Angles

In a polygon, the line running between non-adjacent points is known as a diagonal. The diagonals drawn from a single vertex to the remaining vertices in an n-sided polygon will divide the figure into n-2 triangles. The sum of the interior angles of a convex polygon is then just (n-2)* 180.

If the side of a polygon is extended past the intersecting adjacent side, it defines the exterior angle of the vertex. Each vertex of a convex polygon has two possible exterior angles, defined by the continuation of each of the sides. These two angles are congruent, however, so the exterior angle of a polygon is defined as one of the two angles. The sum of the exterior angles of any convex polygon is equal to 360°.

Kristin Lewotsky

KEY TERMS


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Angle

—A geometric figure created by two lines drawn from the same point.

Concave

—A polygon whose at least one angle is larger than the straight angle (180°).

Convex

—A polygon whose all angles are less than the straight angle (180°).

Diagonal

—The line which links-connects any two non-adjacent vertices.

Equiangular

—A polygon is equiangular if all of its angles are identical.

Equilateral

—A polygon is equilateral if all the sides are equal in length.

Parallelogram

—A rectangle with both pair of sides parallel.

Perimeter

—The sum of the length of all sides.

Rectangle

—A parallelogram in which all angles are right angles.

Reflex polygon

—A polygon in which two non-adjacent sides intersect.

Regular polygon

—An equilateral, equiangular polygon.

Rhombus

—A parallelogram whose adjacent sides are equal.

Square

—A four-sided shape whose sides are equal.

Vertex

—The point at which the two sides of an angle meet.

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