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Topic: Polygon area

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Polygon

Exact trigonometric constants

Decagon

Quadrilateral

Dodecagon

Heptagon

Pentagon

Hexagon

Cartesian coordinate system

American English

	Polygon (Site not responding. Last check: 2007-11-05)
		A polygon (from the Greek poly, for "many", and gwnos, for "angle") is a closed planar path composed of a finite number of sequential straight line segments.
		Polygons are named according to the number of sides, combining a Greek root with the suffix -gon, e.g.
		If the polygon can be drawn on an equally-spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points.
	www.sciencedaily.com /encyclopedia/polygon (671 words)

	efg's Graphics -- Polygon Area and Centroid
		The purpose of this experiment is to demonstrate two ways to compute the area of a polygon, whose coordinates are defined in "world" coordinates.
		The area computed this way is a signed value, where a negative sign indicates the vertices are in clockwise order and a positive sign indicates the vertices are in a counterclockwise order.
		The area is computed from the ratio of the solid color to the total number of pixels times the area in real coordinates.
	www.efg2.com /Lab/Graphics/PolygonArea.htm (685 words)

	Area of Parallelograms
		The area of a polygon is the number of square units inside the polygon.
		The area of a parallelogram is 24 square centimeters and the base is 4 centimeters.
		The area of a parallelogram is 64 square inches and the height is 16 inches.
	www.mathgoodies.com /lessons/vol1/area_parallelogram.html (326 words)

	Polygon area (Site not responding. Last check: 2007-11-05)
		This is a general formula for the area of any polygon; it does not depend on triangles as a special case.
		It depends on the polygon's being simple: that is, dividing the plane into regions with winding numbers of zero and one.
		If a polygon intersects itself, each region of the polygon will contribute to the result a figure of the region's area times its winding number.
	wiki.tcl.tk /12081 (322 words)

	[No title]
		Other approaches can be found4, where the polygon is clipped as a polyline and only the Y intersections are used to generate the different turning points.
		Polygons with multiple boundaries are clipped one boundary at a time, so the proposed method concentrates on the case of a single boundary.
		This guarantees a correctly clipped polygon when all the segments of the polygon are at least partially inside the clipping region, and some end points of the segments are in regions where the code, calculated by the Sutherland-Cohen algorithm has two bits; i.e.
	www.chez.com /pmaillot/2dpclip/2dpclip.htm (3666 words)

	Polygon Area and Centroid (Site not responding. Last check: 2007-11-05)
		This C function returns the area of a polygon.
		The units are the square of the ones used for the vertices of the polygon.
		The problem of determining the area of a polygon seems at best messy but the final formula is particularly simple.
	astronomy.swin.edu.au /~pbourke/geometry/polyarea (418 words)

	Regular Polygon
		The center of a regular polygon is defined as the point inside the polygon which is equidistant to all the vertexes of it.
		The radii of a regular polygon are defined as the lines joining the vertexes and the center.
		Further evidence is the fact that the area of the infinite-sided regular polygon is in fact identical to the area of a circle.
	www.welltall.com /ymc/discovery/polygon.html (513 words)

	web
		You can find the area of a triangle by using this formula 1/2bh= area of triangle, were b is the base of the triangle, and h is the height.
		The polygons AEDB and JIHG have sides of equal length, but their areas are different, with both having a larger area than even the largest area triangle.
		Since there is not a simple equation to solve for the area of polygons, you need to dissect a polygon into shapes that we do have a formula for.
	www.gvsu.edu /math/students/bc/web.htm (808 words)

	Area of Trapezoids Lesson
		To find the area of a trapezoid, take the sum of its bases, multiply the sum by the height of the trapezoid, and then divide the result by 2, The formula for the area of a trapezoid is:
		The area of a trapezoid is 52 square inches and the bases are 11 inches and 15 inches.
		The area of a trapezoid is 48 square inches and the bases are 4 inches and 8 inches.
	www.mathgoodies.com /lessons/vol1/area_trapezoid.html (411 words)

	ATSDR - PHA - FORT ORD, MARINA, MONTEREY COUNTY, COUNTY, CALIFORNIA
		The area is retained by Marina as a buffer at the end of the runway but managed by the University of California Natural Reserve System as a coastal scrub/grassland habitat adjacent to University Research Area.
		This area also boasts excellent bay views available for a combination of multi-family residential and commercial/office/cultural uses, much of which can be integrated to take advantage of pedestrian and transit opportunities as well as the excellent freeway access.
		This area is planned for eventual expansion of not only the university curriculum (e.g., a future environmental studies center) but also possible additional housing units to serve the needs of the students and faculty.
	www.atsdr.cdc.gov /HAC/PHA/fortord/for_p4.html (3976 words)

	Comp.Graphics.Algorithms FAQ, Section 2
		This suggests first triangulating the polygon, then forming a sum of the centroids of each triangle, weighted by the area of each triangle, the whole sum normalized by the total polygon area.
		This indeed works, but there is a simpler method: the triangulation need not be a partition, but rather can use positively and negatively oriented triangles (with positive and negative areas), as is used when computing the area of a polygon.
		Unlike intersections of general polygons, which might produce a quadratic number of pieces, intersection of convex polygons of n and m vertices always produces a polygon of at most (n+m) vertices.
	www.exaflop.org /docs/cgafaq/cga2.html (1062 words)

	Area
		The area of polygon ABCDEF is the area between ABCD and the x axis minus the area between DEFA and the x axis.
		Note that the area between ABCD and the x axis is the sum of the three trapezoids ABB'A', BCC'B', and CDD'C'.
		Similarly, the area between DEFA and the x axis is the sum of three trapezoids.
	www.attewode.com /Calculus/AreaMeasurement/area.htm (1076 words)

	Citations: The area of a simple polygon - Rokne (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		The measurement of the area of the polygon is done analytically on the polygon as opposed to pixel counting.
		....stated previously, virtual environments containing a large number of polygons are expected to be subjects to the spatial subdivision, what gives a good reason to include a brief discussion of the polygon area computation.
		In vector calculus the area of a polygon is often taught of as a vector
	citeseer.ist.psu.edu /context/445206/0 (543 words)

	Analyses supported by IAN (as of October 7th, 2004)
		PA is calculated by averaging the perimeter to area ratio for all polygons present.
		The average corrected perimeter to area ratio is calculated by averaging the corrected perimeter to area ratio for all polygons present.
		Let m be the difference between the area of the polygon and the area of sqr(n).
	landscape.forest.wisc.edu /projects/ian/analyses.htm (2567 words)

	Lecture 9 - Simple algorithms II
		Discuss the results of using the polygon area and point in polygon algorithms when the polygon is not correctly structured (e.g.
		Modify the point in polygon algorithm to determine correctly if the point lies on the boundary of the polygon, in addition to inside or outside.
		Derive the equations for polygon area and centroid from first principles.
	www.geog.ucsb.edu /~good/176b/a08.html (1155 words)

	83.01.07: Geometric Shapes in Architecture
		A polygon is named by placing a capital letter on each vertex, moving consecutively around the figure in either a clockwise or counterclockwise direction.
		The area of each triangle is equal to one half the area of the parallelogram.
		The area of the trapezoid is equal to the sum of the areas of these three polygons.
	www.cis.yale.edu /ynhti/curriculum/units/1983/1/83.01.07.x.html (3540 words)

	Area of Triangles and Polygons in 2D and 3D (Site not responding. Last check: 2007-11-05)
		This computation gives a signed area for a polygon; and, similar to the signed area of a triangle, is positive when the vertices are oriented counterclockwise around the polygon, and negative when oriented clockwise.
		Thus, the 3D planar area can be recovered by a single extra multiplication, and in total this algorithm uses n+5 multiplications, 2n+1 additions, 1 square root (when n is not a unit normal), plus a small overhead choosing the coordinate to ignore.
		We represent a polygon as an array of points, but it is often more convenient to have it as a linked list of vertices (to allow insertion or deletion during drawing operations), and the polygon routines can be easily modified to scan through the linked list (see [O'Rourke, 1998] for an example of this approach).
	geometryalgorithms.com /Archive/algorithm_0101 (2622 words)

	An Improved Algorithm for Calculating the Perimeter and Area of Raster Polygons
		For the area calculation, they typically count the number of pixels that make up the polygon, then multiply that total by the area of one pixel to get the area of the polygon.
		For the perimeter calculation, they typically count the number of edges of individual pixels that are considered on the boundary of the polygon, then multiply that total by the length of the side of the pixel (assuming square pixels) to get the perimeter of the polygon.
		The error is not as severe for area calculations, since only the edge pixels of a polygon contribute to the aliasing error, but it can be as high as 20 percent, if the area of the polygon is small relative to the perimeter.
	www.geovista.psu.edu /sites/geocomp99/Gc99/076/abs99-076.htm (482 words)

	Polygons - Area of polygons and circles - First Glance (Site not responding. Last check: 2007-11-05)
		The area of a shape is a number that tells how many square units are needed to cover the shape.
		Area can be measured in different units, such as square feet, square meters, or square inches.
		Every polygon and circle has a formula for finding its area.
	www.math.com /school/subject3/lessons/S3U2L4GL.html (102 words)

	UNIT 33 - SIMPLE ALGORITHMS II - POLYGONS
		this allows for trapezoids formed by the lower portion of the polygon to be subtracted so that, in the end, the total A is the area of the polygon it self
		note: as with the area algorithm, the polygon must be coded clockwise and all y coordinates must be non- negative
		Modify the polygon area algorithm to test for and accommodate negative y coordinates.
	www.geog.ubc.ca /courses/klink/gis.notes/ncgia/u33.html (1356 words)

	TatukGIS FAQ - DK - Measurements
		Yes, access/retrieval is possible to each point, line segment, or polygon area.
		What about a function to make a hole in the polygon and then calculate the area of the polygon minus the area of the hole?
		The function shape.area returns the area of the polygon minus the area of holes in that polygon.
	www.tatukgis.com /faq/book/bp21.aspx (160 words)

	Area of polygon - GameDev.Net Discussion Forums
		area of a polygon from 3d vertices ie a quad, for use
		Area = (0 + 28 + 0 + 6)/2 = 17.
		The square of the AREAS of the three small sides equal the square of the AREA of the large side, which is diagonal in all 3 directions.
	www.gamedev.net /community/forums/viewreply.asp?ID=307461 (1135 words)

	CSG Irregular Polygon Area Calculator
		It is to determine the area of an irregular polygon.
		The generally accepted manner for finding the area of an irregular polygon is to break it up into triangles and possibly a rectangle, then calculate each and add the totals.
		Other area calculators for your use are our Area Calculator, Ellipse Area Calculator, Parabola Area Calculator, Parallelogram Area Calculator, Pyramid Area Calculator, Trapezoid Area Calculator, our Surface Area Calculator and our Land Parcel Area Calculator.
	www.csgnetwork.com /areairregpolycalc.html (257 words)

	Technical FAQ (1178) How can I calculate the area inside a polygon? (Site not responding. Last check: 2007-11-05)
		You should replace TESTPOLY with a data set containing X and Y coordinates for the polygon whose area you want to calculate (with each set of points in the polygon being a set of X and Y variables).
		The program cannot be used with self-intersecting polygons (that is, a polygon whose boundaries cross each other).
		You will probably want to convert the units to some standard measure, such as miles or meters, before calculating the area.
	support.sas.com /faq/011/FAQ01178.html (202 words)

	Finding the Missing Sides of a Right Triangle
		Definition: An apothem of a regular polygon is any perpendicular line segment drawn from the center of the polygon to a side that forms a right angle with that side.
		Area of a regular polygon is A = ap/2 where A is the area of the polygon, a is the apothem of the polygon, and p is the perimeter of the polygon.
		1) Find the area of a regular octagon whose sides have a length of 10 cm and whose apothem is 5 cm.
	www.scifimath.com /solutions48.htm (144 words)

	Polygon\Region area calculator
		More information: Polygon\Region area calculator It is an application showing a method of calculating the area of any type of polygon (Convex or concave)...It doesn't matter...also, it uses two methods of calculations: 1- Recursive method which has never seen to be used before on the net.
		if u drew any polygon and u found the results of the two methods are different, Plz, save the polygon and send it to me on wael_eng@hotmail.com Also, u can send any comments to enhance my work..I will be pleasant for that....
		Important Issue: What to do to calculate the area of a polygon which can't be drawn on the form due to its size limitations?
	www.freevbcode.com /ShowCode.asp?ID=6962 (503 words)

	WSJ.com - Dying Mathematician Spends Last Days on Area of Polygon
		If the polygon happens to be a rectangle, the answer is easy: Multiply the height by the width to get the area.
		For everyday purposes, finding the area of a seven-sided figure is easy: One just divides the figure into triangles, physically measures the sides of the triangles and then uses the ancient formula to calculate the area of each triangle.
		When he was 12 or 13, a discussion with his father about triangles led Dr. Robbins to figure out the general solution for the area of triangles.
	www.plambeck.org /oldhtml/mathematics/robbins.htm (1288 words)

	AREA
		AREA SYMBOLS ARE APPLIED TO THE CHOROGRAMS ACCORDING TO THE VALUES IN EACH UNIT AND THE SYMBOLS CHOSEN TO REPRESENT THEM.
		SOMETIMES THE COMPARISONS OF DIFFERENT-SIZED POLYGON AREAS OR THEIR PERIMETERS IS OF INTEREST IN ITSELF.
		IN THIS KIND OF SERIES THE AREA ON THE MAP IS DIVIDED INTO EQUAL REGIONS.
	coss.stcloudstate.edu /brichason/area.htm (1361 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Here is a lightly-edited version of a post I once made showing how to compute not only the area but the center of mass.
		At the end, the counter Area contains the area of the polygon; the center of mass is at (Xcoord/Area, Ycoord/Area).
		Moreover, I should be careful to state that this gives the _signed_ area: in Green's theorem it is assumed that you walk around the curve with the interior on your left.
	www.math.niu.edu /~rusin/known-math/95/greens (595 words)

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