HANDOUTS ON POLYGONS (KINDS AND FORMS)

PREPARED BY: LOURISE ARCHIE C. SUBANG

A polygon can be defined (as illustrated above) as a geometric object "consisting of a number of points (called vertices) and an equal number of line segments (called sides), namely a cyclically ordered set of points in a plane, with no three successive points collinear, together with the line segments joining consecutive pairs of the points. In other words, a polygon is closed broken line lying in a plane."

If all sides and angles are equivalent, the polygon is called regular. Polygons can be convex, concave, or star. The word "polygon" derives from the Greek poly, meaning "many," and gonia, meaning "angle."

The most familiar type of polygon is the regular polygon, which is a convex polygon with equal sides, lengths and angles.

PRINCIPAL PARTS OF THE POLYGON SIDE

Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral or quadrangle, and nonagon are exceptions. For large numbers, mathematicians usually write the numeral itself, e.g. 17-gon. A variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula.

Polygon names

Name Edges Remarks

henagon (or monogon) 1 In the Euclidean plane, degenerates to a closed curve with a single vertex point on it.

digon 2 In the Euclidean plane, degenerates to a closed curve with two vertex points on it.

triangle (or trigon) 3 The simplest polygon which can exist in the Euclidean plane.

quadrilateral (or quadrangle or tetragon)

4 The simplest polygon which can cross itself.

pentagon 5 The simplest polygon which can exist as a regular star. A star pentagon is known as a pentagram or pentacle.

hexagon 6 avoid "sexagon" = Latin [sex-] + Greek

heptagon 7 avoid "septagon" = Latin [sept-] + Greek

octagon 8

enneagon or nonagon 9 "nonagon" is commonly used but mixes Latin [novem = 9] with Greek. Some modern authors prefer "enneagon".

decagon 10

ANGLE Any polygon, regular or irregular, self-intersecting or simple, has as many corners as it has sides. Each corner has several angles. The two most important ones are:

Interior angle – The sum of the interior angles of a simple n-gon is (n − 2)π radians or (n − 2)180 degrees. This is because any simple n-gon can be considered to be made up of (n − 2) triangles, each of which has an angle sum of π radians or 180 degrees. The

measure of any interior angle of a convex regular n-gon is radians or

degrees. The interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra.

Exterior angle – Imagine walking around a simple n-gon marked on the floor. The amount you "turn" at a corner is the exterior or external angle. Walking all the way round the polygon, you make one full turn, so the sum of the exterior angles must be 360°. Moving around an n-gon in general, the sum of the exterior angles (the total amount one "turns" at the vertices) can be any integer multiple d of 360°, e.g. 720° for a pentagram and 0° for an angular "eight", where d is the density or starriness of the polygon. See also orbit (dynamics).

The exterior angle is the supplementary angle to the interior angle. From this the sum of the interior angles can be easily confirmed, even if some interior angles are more than 180°: going clockwise around, it means that one sometime turns left instead of right, which is counted as turning a negative amount. (Thus we consider something like the winding number of the orientation of the sides, where at every vertex the contribution is between −½ and ½ winding.)

TYPES OF POLYGONS ACCORING TO NUMBER OF SIDES CIRLCE

A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point, the centre.

Circles are simple closed curves which divide the plane into two regions, an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.

A circle is a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.

TRIANGLES A triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. Every triangle has three sides and three angles, some of which may be the same. The sides of a triangle are given special names in the case of a right triangle, with the side opposite the right angle being termed the hypotenuse and the other two sides being known as the legs. All triangles are convex and bicentric. That portion of the plane enclosed by the triangle is called the triangle interior, while the remainder is the exterior. The study of triangles is sometimes known as triangle geometry, and is a rich area of geometry filled with beautiful results and unexpected connections. In 1816, while studying the Brocard points of a triangle, Crelle exclaimed, "It is indeed wonderful that so simple a figure as the triangle is so inexhaustible in properties. How many as yet unknown properties of other figures may there not be?" (Wells 1991, p. 21).

It is common to label the vertices of a triangle in counterclockwise order as either , , (or

, , ). The vertex angles are then given the same symbols as the vertices themselves. The

symbols , , (or , , ) are also sometimes used (e.g., Johnson 1929), but this convention results in unnecessary confusion with the common notation for trilinear coordinates , and so is not recommended. The sides opposite the angles , , and (or

, , ) are then labeled , , (or , , ), with these symbols also indicating the lengths of the sides (just as the symbols at the vertices indicate the vertices themselves as well as the vertex angles, depending on context).

An triangle is said to be acute if all three of its angles are all acute, a triangle having an obtuse angle is called an obtuse triangle, and a triangle with a right angle is called right. A triangle with all sides equal is called equilateral, a triangle with two sides equal is called isosceles, and a triangle with all sides a different length is called scalene. A triangle can be simultaneously right and isosceles, in which case it is known as an isosceles right triangle

QUADRILATERALS

A parallelogram is a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Parallelograms also include the square, rectangle, rhombus and rhomboid.

TRAPEZIUM is a quadrilateral with no parallel sides (a shape known elsewhere as a general irregular quadrilateral).

In geometry, a quadrilateral with one pair of parallel sides is referred to as a trapezoid. Some authors[2] define a trapezoid as a quadrilateral having exactly one pair of parallel sides, thereby excluding parallelograms. Other authors[3] define a trapezoid as a quadrilateral with at least one pair of parallel sides, making the parallelogram a special type of trapezoid (along with the rhombus, the rectangle and the square). The latter definition is consistent with its uses in higher mathematics such as calculus.

Rhombus: all four sides are of equal length. Equivalent conditions are that opposite sides are parallel and opposite angles are equal, or that the diagonals perpendicularly bisect each other. An informal description is "a pushed-over square" (including a square).

Rectangle: all four angles are right angles. An equivalent condition is that the diagonals bisect each other and are equal in length. Informally: "a box or oblong" (including a square).

Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. An equivalent condition is that opposite sides are parallel (a square is a parallelogram), that the diagonals perpendicularly bisect each other, and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle (four equal sides and four equal angles).

PENTAGON

In geometry, a pentagon (from pente, which is Greek for the number 5) is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon. A regular pentagon has all sides of equal length and all interior angles are equal measure (108°). It has five lines of reflectional symmetry and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). Its Schläfli symbol is {5}. The chords of a regular pentagon are in golden ratio to its sides.

HEXAGON

A regular hexagon has all sides of the same length, and all internal angles are 120°. A regular hexagon has 6 rotational symmetries (rotational symmetry of order six) and 6 reflection symmetries (six lines of symmetry), making up the dihedral group D6. The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side.

Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane (three hexagons meeting at every vertex), and so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons.

HEPTAGON

In geometry, a heptagon (or septagon) is a polygon with seven sides and seven angles. In a regular heptagon, in which all sides and all angles are equal, the sides meet at an angle of 5π/7 radians, 128.5714286 degrees. Its Schläfli symbol is {7}. The area (A) of a regular heptagon of side length a is given by

The heptagon is also 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). The OED lists "septagon" as meaning "heptagonal".

OCTAGON

In geometry, an octagon (from the Greek okto, eight[1]) is a polygon that has eight sides. A regular octagon is represented by the Schläfli symbol {8}. A regular octagon is a closed figure with sides of the same length and internal angles of the same size. It has eight lines of reflective symmetry and rotational symmetry of order 8. The internal angle at each vertex of a regular octagon is 135° and the sum of all the internal angles is 1080° (as for any octagon).

NONAGON

In geometry, a nonagon (or enneagon) is a nine-sided polygon.

The name "nonagon" is a prefix hybrid formation, from Latin (nonus, "ninth" + gonon), used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century. The name "enneagon" comes from Greek enneagonon (εννεα, nine + γωνον (from γωνία = corner)), and is arguably more correct, though somewhat less common.

A regular nonagon has internal angles of 140°. The area of a regular nonagon of side length a is given by

Although a regular nonagon is not constructible with compass and straightedge there are methods of construction that produce very close approximations.

DECAGON

In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and each internal angle equal to 144°. Its Schläfli symbol is {10}.

TYPES OF POLGON ACCORDING TO THE MEASURES OF ANGLES AND SIDES

REGULAR POLYGONS

A regular polygon is a polygon which is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be convex or star. A regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle (the circumscribed circle), i.e., they are concyclic points. Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle. A regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon.

IRREGULAR POLYGONS An irregular polygon is a polygon whose characteristics are opposite to the regular polygons.

REFERENCES: http://en.wikipedia.org/wiki/Irregular_polygon teachnology.com google.com

excite.com teachnet.com EXPLORING MATHEMATICS – GEOMETRY, ORLANDO A. ORONCE