From Latin: planus "flat, level," and Greek: geometrical "measurement of earth or land"The study of geometry can be broken into two broad types: plane geometry, which deals with only two dimensions, and and solid geometry which allows all three. The world around us is obviously three-dimensional, having width, depth and height, Solid geometry deals with objects in that space such as cubes and spheres.Plane geometry deals in objects that are flat, such as triangles and lines, that can be drawn on a flat piece of paper.

The PlaneIn plane geometry, all the shapes exist in a flat plane. A plane can be thought of an a flat sheet with no thickness, and which goes on for ever in both directions. It is absolutely flat and infinitely large, which makes it hard to draw. In the figure above, the yellow area is meant to represent a plane. In the figure, it has edges, but actually, a plane goes on for ever in both directions.Objects which lie in the same plane are said to be 'coplanar'. SeeDefintion of coplanar.OriginsPlane geometry, and much of solid geometry also, was first laid out by the Greeks some 2000 years ago.Euclidin particular made great contributions to the field with his book "Elements" which was the first deep, methodical treatise on the subject. In particular, he built a layer-by-layer sequence of logical steps, proving beyond doubt that each step followed logically from those before.Geometry is really about two things:1. The objects and their properties. Analysis of things such as points, lines, triangles.2. Proofs. A methodology for proving that the claims made about objects are really true.Fun readingClearly, our world is three dimensional. But in the fictional storyFlatlandby Edwin Abbott, he speculates what living in a two-dimensional world (a plane) would be like. It's a fun diversion from the strict factual logic of mathematics. Surprisingly for a science fiction story, it was written in 1884, and his writing style is quaintly Victorian as a result. An excerpt from Chapter 1:..Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows ...Read it online at "Flatland - A romance of many dimensions" by Edwin A. Abbott, a Square,1884;Other line topicsDefinition: A shape, formed by two lines or rays diverging from a common point (the vertex).Try thisAdjust the angle below by dragging the orange dot.AttributesVertexThevertexis the common point at which the two lines or rays are joined. Point B is the figure above is the vertex of the angleABC.

LegsThe legs (sides) of an angle are the two lines that make it up. In the figure above, the line segments AB and BC are the legs of the angleABC.

InteriorThe interior of an angle is the space in the 'jaws' of the angle extending out to infinity. SeeInterior of an Angle

ExteriorAll the space on the plane that is not the interior. SeeInterior of an Angle

Identifying an angleAn angle can be identified in two ways.1. Like this:ABCThe angle symbol, followed by three points that define the angle, with the middle letter being the vertex, and the other two on the legs. So in the figure above the angle would beABC orCBA. So long as the vertex is the middle letter, the order is not important. As a shorthand we can use the 'angle' symbol. For example 'ABC' would be read as 'the angle ABC'.2. Or like this:BJust by the vertex, so long as it is not ambiguous. So in the figure above the angle could also be called simply 'B'Measure of an angleThe size of an angle is measured in degrees (seeAngle Measures). When we say 'the angle ABC' we mean the actual angle object. If we want to talk about the size, or measure, of the angle in degrees, we should say 'the measure of the angle ABC' - often written mABC.

However, many times we will see 'ABC=34'. Strictly speaking this is an error. It should say 'mABC=34'Types of angleAltogether, there are six types of angle as listed below. Click on an image for a full description of that type and a corresponding interactive applet.

Acute angleLess than 90Right angleExactly 90Obtuse angleBetween 90 and 180

Straight angleExactly 180Reflex angleBetween 180 and 360Full angleExactly 360

In TrigonometryWhen used intrigonometry, angles have some extra properties: They can have a measure greater than 360, can be positive and negative, and are positioned on a coordinate grid with x and y axes. They are usually measured inradiansinstead ofdegrees. For more on this seeAngle definition and properties (trigonometry).Angle constructionIn the Constructions chapter, there are animated demonstrations of variousconstructionsof angles using only a compass and straightedge.

Copying an angle Constructing a 30 angle Constructing a 45 angle Constructing a 60 angle Constructing a 90 angle (perpendicular, right angle) at: the end of a line segment a point on a line segment through a point not on a line segment the midpoint of a a line segment Definition: A shape, formed by two lines or rays diverging from a common point (the vertex). Try thisAdjust the angle below by dragging the orange dot. What kind of angle is shown in each of the diagrams below? a)

b)

c)

d)

e)

f)

Area of a polygon(Coordinate Geometry)A method for finding the area of any polygon when thecoordinatesof itsverticesare known.(See also:Computer algorithm for finding the area of any polygon.)First, number the vertices in order, going either clockwise or counter-clockwise, starting at any vertex.

The area is then given by the formula

Where x1is the x coordinate of vertex 1 and ynis the y coordinate of the nth vertex etc. Notice that the in the last term, the expression wraps around back to the first vertex again.Try it hereAdjust the quadrilateral ABCD by dragging any vertex. The area is calculated using this method as you drag. A detailed explanation follows the diagram.The above diagram shows how to do this manually.1. Make a table with the x,y coordinates of each vertex. Start at any vertex and go around the polygon in either direction. Add the starting vertex again at the end. You should get a table that looks like the leftmost gray box in the figure above.2. Combine the first two rows by:1. Multiplying the first row x by the second row y. (red)2. Multiplying the first row y by the second row x (blue)3. Subtract the second product form the first.3. Repeat this for rows 2 and 3, then rows 3 and 4 and so on.4. Add these results, make it positive if required, and divide by two.Area calculatorSeePolygon area calculatorfor a pre-programmed calculator that does the arithmetic for you. Just enter the coordinates.LimitationsThis method will produce the wrong answer for self-intersecting polygons, where one side crosses over another, as shown on the right. Itwillwork correctly however fortriangles,regularandirregular polygons,convexorconcave polygons.Things to tryIn the above diagram, press 'reset' and 'hide details', then try the following:1. Drag the vertices of the polygon to create a new shape. (Do not create a 'crossed' polygon, this method does not work on those.)2. Calculate the area using this method.3. Click on 'show details' to check your answer.

quadrilateral where all interior angles are 90, and whose location on thecoordinate planeis deteampleThe example below assumes you know how to calculate the distance between two points, as described inDistance between Two Points. In the figure above, click 'reset' and 'show diagonals' The heightof the rectangle is the distance between the points A and B. (Using C,D will produce the same result). Using the formula for the distance between two points, this isCalculator

The widthis the distance between the points B and C. (Using A,D will produce the same result). Using the formula for the distance between two points, this is The length of a diagonalsis the distance between B and D. (Using A,C will produce the same result). Using the formula for the distance between two points, this isrmined by thecoordinatesof the fourvertices(corners).

The example below assumes you know how to calculate the distance between two points, as described inDistance between Two Points. In the figure above, click 'reset', 'rotated' and 'show diagonals' The side lengthof the square is the distance between any two adjacentvertices. Let's pick B and C. Using the formula for the distance between two points:

The length of a diagonalsis the distance between any pair of opposite vertices. In a square, the diagonal is also the length of a side times the square root of two:Calculator

Square(Coordinate Geometry)A 4-sided regularpolygonwith all sides equal, all interior angles 90 and whose location on thecoordinate planeis determined by thecoordinatesof the fourvertices(corners).Try thisDrag any vertex of the square below. It will remain a square and its dimensions calculated from its coordinates. You can also drag the origin point at (0,0), or drag the square itself.In coordinate geometry, a square is similar to an ordinary square (SeeSquare definition )with the addition that its position on thecoordinate planeis known. Each of the four vertices (corners) have knowncoordinates. From these coordinates, various properties such as width, height etc can be found.It has all the same properties as a familiar square, such as: All four sides arecongruent Opposite sides areparallel The diagonalsbisecteach other at right angles The diagonals arecongruentSeeSquare definitionfor more.\

rapezoid(Coordinate Geometry)Aquadrilateralthat has one pair of parallel sides,and where theverticeshave knowncoordinates.Try thisDrag any vertex of the trapezoid below. It will remain a trapezoid. You can also drag the origin point at (0,0).As in plane geometry, a trapezoid is aquadrilateralwith one pair of parallel sides. (SeeTrapezoid definition). In coordinate geometry, each of the four vertices (corners) also have knowncoordinates.Altitude of a trapezoidIn the figure above, click on 'reset' then 'show altitude'. The altitude is the perpendicular distance between the two bases (parallel sides). To find this distance, we can use the methods described inDistance from a point to a line. For the point, we use any vertex, and for the line we use the opposite base. In the figure above we have used the distance from point B to the opposite base AD.This method will work even if the trapezoid is rotated on the plane, but if the sides of the trapezoid are parallel to the x and y axes, then the calculations can be a little easier. The altitude is then the difference in y-coordinates of any point on each base, for example A and B.

rallelogram(Coordinate Geometry)Aquadrilateralwith both pairs of opposite sides parallel andcongruent, and whose location on thecoordinate planeis determined by thecoordinatesof the fourvertices(corners).Try thisDrag any vertex of the parallelogram below. It will remain a parallelogram and its dimensions calculated from its coordinates. You can also drag the origin point at (0,0).In coordinate geometry, a parallelogram is similar to an ordinary parallelogram (Seeparallelogram definition )with the addition that its position on thecoordinate planeis known. Each of the four vertices (corners) have knowncoordinates. From these coordinates, various properties such as its altitude can be found.It has all the same properties as a familiar parallelogram: Opposite sides are parallel and congruent The diagonals bisect each other Opposite angles are congruentSeeparallelogram definitionfor more.Dimensions of a parallelogramThe dimensions of the parallelogram are found by calculating the distance between various corner points. Recall that we can find the distance between any two points if we know their coordinates. (SeeDistance between Two Points) So in the figure above: The heightof the parallelogram is the distance between A and B (or C,D). The widthis the distance between B and C (or A,D). The length of adiagonalsis the distance between opposite corners, say B and D (or A,C since the diagonals are congruent).This method will work even if the parallelogram is rotated on the plane, as in the figure above. But if the sides of the parallelogram are parallel to the x and y axes, then the calculations can be a little easier.In the above figure uncheck the "rotated" box to create this condition and note that: The heightis the difference in y-coordinates of any top and bottom point - for example A and B. The widthis the difference in x-coordinates of any left and right point - for example B and DExampleThe example below assumes you know how to calculate the distance between two points, as described inDistance between Two Points. In the figure above, click 'reset' and 'show diagonals' The heightof the parallelogram is the distance between the points A and B. (Using C,D will produce the same result). Using the formula for the distance between two points, this is The widthis the distance between the points B and C. (Using A,D will produce the same result). Using the formula for the distance between two points, this is The length of a diagonalsis the distance between B and D. (Using A,C will produce the same result). Using the formula for the distance between two points, this is