Regular Polygon free essay sample

Geometrically two edges meeting at a corner are required to form an angle that is not straight (180Â °); otherwise, the line segments will be considered parts of a single edge – however mathematically, such corners may sometimes be allowed. In fields relating to computation, the term polygon has taken on a slightly altered meaning derived from the way the shape is stored and manipulated in computer graphics (image generation). Some other generalizations of polygons are described below. Polygons are primarily classified by the number of sides. See table below. Convexity and types of non-convexity Polygons may be characterized by their convexity or type of non-convexity: * Convex: any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. Equivalently, all its interior angles are less than 180Â °. * Non-convex: a line may be found which meets its boundary more than twice. In other words, it contains at least one interior angle with a measure larger than 180Â °. * Simple: the boundary of the polygon does not cross itself. All convex polygons are simple. * Concave: Non-convex and simple. * Star-shaped: the whole interior is visible from a single point, without crossing any edge. The polygon must be simple, and may be convex or concave. * Self-intersecting: the boundary of the polygon crosses itself. Branko Grunbaum calls these coptic, though this term does not seem to be widely used. The term complex is sometimes used in contrast to simple, but this risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert plane consisting of two complex dimensions. Star polygon: a polygon which self-intersects in a regular way. Symmetry * Equiangular: all its corner angles are equal. * Cyclic: all corners lie on a single circle. * Isogonal or vertex-transitive: all corners lie within the same symmetry orbit. The polygon is also cyclic and equiangular. Equilateral: all edges are of the same length. (A polygon with 5 or more sides can be equilateral without being convex. ) [1] * Isotoxal or edge-transitive: all sides lie within the same symmetry orbit. The polygon is also equilateral. * Tangential: all sides are tangent to an inscribed circle. Regular: A polygon is regular if it is both cyclic and equilateral. A non-convex regular polygon is called a regular star polygon. Miscellaneous * Rectilinear: a polygon whose sides meet at right angles, i. e. , all its interior angles are 90 or 270 degrees. * Monotone with respect to a given line L, if every line orthogonal to L intersects the polygon not more than twice. Properties Euclidean geometry is assumed throughout. Angles 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)? 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 ? 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 – Tracing around a convex n-gon, the angle turned at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360Â °. This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an n-gon in general, the sum of the exterior angles (the total amount one rotates at the vertices) can be any integer multiple d of 360Â °, e. g. 20Â ° for a pentagram and 0Â ° 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 180Â °: going clockwise around, it means that one sometime turns left instead of right, which is counted as turning a negative amount. The area formula is derived by taking each edge AB, and calculating the (signed) area of triangle ABO with a vertex at the origin O, by taking the cross-product (which gives the area of a parallelogram) and dividing by 2. As one wraps around the polygon, these triangles with positive and negative area will overlap, and the areas between the origin and the polygon will be cancelled out and sum to 0, while only the area inside the reference triangle remains. This is why the formula is called the Surveyors Formula, since the surveyor is at the origin; if going counterclockwise, positive area is added when going from left to right and negative area is added when going from right to left, from the perspective of the origin. The formula was described by Meister[citation needed] in 1769 and by Gauss in 1795. It can be verified by dividing the polygon into triangles, but it can also be seen as a special case of Greens theorem. Â  The formula is The formula was described by Lopshits in 1963. [3] If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Picks theorem gives a simple formula for the polygons area based on the numbers of interior and boundary grid points. In every polygon with perimeter p and area A , the isoperimetric inequality holds. [4] If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon. This is the Bolyai-Gerwien theorem. The area of a regular polygon is also given in terms of the radius r of its inscribed circle and its perimeter p by . This radius is also termed its apothem and is often represented as a. The area of a regular n-gon with side s inscribed in a unit circle is . The area of a regular n-gon in terms of the radius r of its circumscribed circle and its perimeter p is given by . The area of a regular n-gon, inscribed in a unit-radius circle, with side s and interior angle ? can also be expressed trigonometrically as . The sides of a polygon do not in general determine the area. 5] However, if the polygon is cyclic the sides do determine the area. Of all n-gons with given sides, the one with the largest area is cyclic. Of all n-gons with a given perimeter, the one with the largest area is regular (and therefore cyclic). [6] Self-intersecting polygons The area of a self-intersecting polygon can be defined in two different ways, each of which gives a different answer: * Using the above methods for simple polygons, we discover that particular regions within the polygon may have their area multiplied by a factor which we call the density of the region. For example the central convex pentagon in the center of a pentagram has density 2. The two triangular regions of a cross-quadrilateral (like a figure 8) have opposite-signed densities, and adding their areas together can give a total area of zero for the whole figure. * Considering the enclosed regions as point sets, we can find the area of the enclosed point set. This corresponds to the area of the plane covered by the polygon, or to the area of a simple polygon having the same outline as the self-intersecting one (or, in the case of the cross-quadrilateral, the two simple triangles). Degrees of freedom An n-gon has 2n degrees of freedom, including 2 for position, 1 for rotational orientation, and 1 for overall size, so 2n ? 4 for shape. In the case of a line of symmetry the latter reduces to n ? 2. Let k ? 2. For an nk-gon with k-fold rotational symmetry (Ck), there are 2n ? 2 degrees of freedom for the shape. With additional mirror-image symmetry (Dk) there are n ? 1 degrees of freedom. Product of distances from a vertex to other vertices of a regular polygon For a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. Generalizations of polygons In a broad sense, a polygon is an unbounded (without ends) sequence or circuit of alternating segments (sides) and angles (corners). An ordinary polygon is unbounded because the sequence closes back in itself in a loop or circuit, while an apeirogon (infinite polygon) is unbounded because it goes on for ever so you can never reach any bounding end point. The modern mathematical understanding is to describe such a structural sequence in terms of an abstract polygon which is a partially ordered set (poset) of elements. The interior (body) of the polygon is another element, and (for technical reasons) so is the null polytope or nullitope. A geometric polygon is understood to be a realization of the associated abstract polygon; this involves some mapping of elements from the abstract to the geometric. Such a polygon does not have to lie in a plane, or have straight sides, or enclose an area, and individual elements can overlap or even coincide. For example a spherical polygon is drawn on the surface of a sphere, and its sides are arcs of great circles. So when we talk about polygons we must be careful to explain what kind we are talking about. A digon is a closed polygon having two sides and two corners. On the sphere, we can mark two opposing points (like the North and South poles) and join them by half a great circle. Add another arc of a different great circle and you have a digon. Tile the sphere with digons and you have a polyhedron called a hosohedron. Take just one great circle instead, run it all the way around, and add just one corner point, and you have a monogon or henagon – although many authorities do not regard this as a proper polygon. Other realizations of these polygons are possible on other surfaces, but in the Euclidean (flat) plane, their bodies cannot be sensibly realized and we think of them as degenerate. The idea of a polygon has been generalized in various ways. Here is a short list of some degenerate cases (or special cases, depending on your point of view): * Digon: Interior angle of 0Â ° in the Euclidean plane. See remarks above re. on the sphere. * Interior angle of 180Â °: In the plane this gives an apeirogon (see below), on the sphere a dihedron * A skew polygon does not lie in a flat plane, but zigzags in three (or more) dimensions. The Petrie polygons of the regular polyhedra are classic examples. * A spherical polygon is a circuit of sides and corners on the surface of a sphere.