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Handouts on polygons


a summarized handouts of the basic concepts of polygons.

a summarized handouts of the basic concepts of polygons.

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  • 1. 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 anumber of points (called vertices) and an equal number of line segments (called sides), namelya cyclically ordered set of points in a plane, with no three successive points collinear, togetherwith the line segments joining consecutive pairs of the points. In other words, a polygon isclosed broken line lying in a plane." If all sides and angles are equivalent, the polygon is called regular. Polygons can beconvex, 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 withequal sides, lengths and angles.PRINCIPAL PARTS OF THE POLYGONSIDE Individual polygons are named (and sometimes classified) according to the number ofsides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon,dodecagon. The triangle, quadrilateral or quadrangle, and nonagon are exceptions. For largenumbers, mathematicians usually write the numeral itself, e.g. 17-gon. A variable can even beused, usually n-gon. This is useful if the number of sides is used in a formula.Polygon namesName Edges Remarkshenagon (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 4 The simplest polygon which can cross itself.quadrangle or tetragon)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-] + Greekheptagon 7 avoid "septagon" = Latin [sept-] + Greekoctagon 8enneagon or nonagon 9 "nonagon" is commonly used but mixes Latin [novem = 9] with Greek. Some modern authors prefer "enneagon".
  • 2. decagon 10ANGLEAny 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 theinterior angles can be easily confirmed, even if some interior angles are more than 180°: goingclockwise around, it means that one sometime turns left instead of right, which is counted asturning a negative amount. (Thus we consider something like the winding number of theorientation of the sides, where at every vertex the contribution is between −½ and ½ winding.) TYPES OF POLYGONS ACCORING TO NUMBER OF SIDESCIRLCEA circle is a simple shape of Euclidean geometry consisting of those points in a plane which areequidistant from a given point, the centre.Circles are simple closed curves which divide the plane into two regions, an interior and anexterior. In everyday use, the term "circle" may be used interchangeably to refer to either theboundary of the figure, or to the whole figure including its interior; in strict technical usage, thecircle is the former and the latter is called a disk.
  • 3. A circle is a special ellipse in which the two foci are coincident and the eccentricity is 0. Circlesare conic sections attained when a right circular cone is intersected with a plane perpendicularto the axis of the cone.TRIANGLESA triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. Everytriangle has three sides and three angles, some of which may be the same. The sides of atriangle are given special names in the case of a right triangle, with the side opposite the rightangle being termed the hypotenuse and the other two sides being known as the legs. Alltriangles are convex and bicentric. That portion of the plane enclosed by the triangle is calledthe triangle interior, while the remainder is the exterior.The study of triangles is sometimes known as triangle geometry, and is a rich area of geometryfilled with beautiful results and unexpected connections. In 1816, while studying the Brocardpoints of a triangle, Crelle exclaimed, "It is indeed wonderful that so simple a figure as thetriangle is so inexhaustible in properties. How many as yet unknown properties of other figuresmay 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. Thesymbols , , (or , , ) are also sometimes used (e.g., Johnson 1929), but thisconvention results in unnecessary confusion with the common notation for trilinearcoordinates , and so is not recommended. The sides opposite the angles , , and (or , , ) are then labeled , , (or , , ), with these symbols also indicating the lengthsof the sides (just as the symbols at the vertices indicate the vertices themselves as well as thevertex angles, depending on context).An triangle is said to be acute if all three of its angles are all acute, a triangle having an obtuseangle is called an obtuse triangle, and a triangle with a right angle is called right. A triangle withall sides equal is called equilateral, a triangle with two sides equal is called isosceles, and atriangle with all sides a different length is called scalene. A triangle can be simultaneously rightand isosceles, in which case it is known as an isosceles right triangle
  • 4. QUADRILATERALSA parallelogram is a quadrilateral with two pairs of parallel sides. Equivalent conditions are thatopposite sides are of equal length; that opposite angles are equal; or that the diagonals bisecteach 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).PENTAGONIn geometry, a pentagon (from pente, which is Greek for the number 5) is any five-sidedpolygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in asimple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon. A regularpentagon has all sides of equal length and all interior angles are equal measure (108°). It hasfive 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 itssides.HEXAGONA regular hexagon has all sides of the same length, and all internal angles are 120°. A regularhexagon has 6 rotational symmetries (rotational symmetry of order six) and 6 reflectionsymmetries (six lines of symmetry), making up the dihedral group D6. The longest diagonals of aregular hexagon, connecting diametrically opposite vertices, are twice the length of one side.
  • 5. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile theplane (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 makesefficient use of space and building materials. The Voronoi diagram of a regular triangular latticeis the honeycomb tessellation of hexagons.HEPTAGONIn geometry, a heptagon (or septagon) is a polygon with seven sides and seven angles. In aregular heptagon, in which all sides and all angles are equal, the sides meet at an angle of 5π/7radians, 128.5714286 degrees. Its Schläfli symbol is {7}. The area (A) of a regular heptagon ofside length a is given byThe 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). TheOED lists "septagon" as meaning "heptagonal".OCTAGONIn geometry, an octagon (from the Greek okto, eight[1]) is a polygon that has eight sides. Aregular octagon is represented by the Schläfli symbol {8}. A regular octagon is a closed figurewith sides of the same length and internal angles of the same size. It has eight lines of reflectivesymmetry and rotational symmetry of order 8. The internal angle at each vertex of a regularoctagon is 135° and the sum of all the internal angles is 1080° (as for any octagon).NONAGONIn geometry, a nonagon (or enneagon) is a nine-sided polygon.The name "nonagon" is a prefix hybrid formation, from Latin (nonus, "ninth" + gonon), usedequivalently, attested already in the 16th century in French nonogone and in English from the17th 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 isgiven by
  • 6. Although a regular nonagon is not constructible with compass and straightedge there aremethods of construction that produce very close approximations.DECAGON In geometry, a decagon is any polygon with ten sides and ten angles, and usually refersto a regular decagon, having all sides of equal length and each internal angle equal to 144°. ItsSchläfli symbol is {10}. TYPES OF POLGON ACCORDING TO THE MEASURES OF ANGLES AND SIDESREGULAR POLYGONSA regular polygon is a polygon which is equiangular (all angles are equal in measure) andequilateral (all sides have the same length). Regular polygons may be convex or star. A regularn-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on acommon circle (the circumscribed circle), i.e., they are concyclic points. Together with theproperty of equal-length sides, this implies that every regular polygon also has an inscribedcircle or incircle. A regular n-sided polygon can be constructed with compass and straightedge ifand only if the odd prime factors of n are distinct Fermat primes. See constructible polygon.IRREGULAR POLYGONSAn irregular polygon is a polygon whose characteristics are opposite to the regular polygons.REFERENCES: excite.comygon EXPLORING MATHEMATICS – GEOMETRY, ORLANDO A. ORONCE