Geometric figures


Armenian state pedagogical university after khachatur abovyan

Faculty of Mathematics, Physics and Informatics

: Geometric figures

Piruza MkhitaryanSupervisor:. Baghdasaryan


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Geometry is a branch of mathematics <#"justify">There are basic geometric figures that are accepted without definition and are used for defining other concepts, such as point, line, plan and space concepts. In addition to this there are geometric objects, which are widely used in mathematics, physics and other fields. Geometric objects can be two-dimensional, three-dimensional and n-dimensional. Well explore two-dimensional objects.

Two- dimensional objects are called geometric figures, which are located on the plane. Examples of the geometric images are triangles, quadrilateral <#"justify">Three-dimensional objects are located in the space. Examples of three-dimensional objects are the cube, tetrahedron, prism, cone, cylinder, sphere, etc.dimensional and three-dimensional objects also are called geometric shapes.this study I want to introduce some properties of two-dimensional geometric shapes or geometric figures and interesting facts about them.

1. Polygons

polygon can be defined as a geometric object "consisting of a number of points (called vertices) and an equal number of line segments (called sides). In other words, a polygon is closed broken line lying in a plane"

A polygon with vertices (and sides) is known as an -gon. A polygon for which the only points of the plane belonging to two polygon edges of are the polygon vertices is said to be a simple polygon <#"41" src="doc_zip5.jpg" /> sides. The words for polygons with sides (e.g., pentagon <#"41" src="doc_zip7.jpg" />-gon" explicitly. For some polygons, several different terms are used interchangeably, e.g., nonagon and enneagon both refer to the polygon with sides.

. Triangles

A triangle is a 3-sided polygon <#"justify">polygonnpolygon2digon <#"110" src="doc_zip10.jpg" />

is common to label the vertices of a triangle in counterclockwise order as either , (or). The vertex angles <#"25" src="doc_zip13.jpg" /> (or) are also sometimes used, but this convention results in unnecessary confusion with the common notation for trilinear coordinates <#"14" src="doc_zip15.jpg" />, and so is not recommended. The sides opposite the angles(are then labeled with these symbols also indicating the lengths of the sides.are different types of triangles.

Triangles can be classified according to the relative lengths of their sides:

·In an equilateral triangle <#"justify"> <#"justify" height="82" src="doc_zip20.jpg" /> <#"justify" height="70" src="doc_zip21.jpg" /> <#"justify">EquilateralIsoscelesScalenediagrams representing triangles (and other geometric figures), "tick" marks along the sides are used to denote sides of equal lengths - the equilateral triangle has tick marks on all 3 sides, the isosceles on 2 sides. The scalene has single, double, and triple tick marks, indicating that no sides are equal. Similarly, arcs on the inside of the vertices are used to indicate equal angles. The equilateral triangle indicates all 3 angles are equal; the isosceles shows 2 identical angles. The scalene indicates by 1, 2, and 3 arcs that no angles are equal.

Triangles can also be classified according to their internal angles <#"justify">There is interesting fact about right triangle: for every two natural numbers numbers are sides of right triangle.

·Triangles that do not have an angle that measures 90° are called oblique triangles.

·A triangle that has all interior angles measuring less than 90° is an acute triangle or acute-angled triangle.

·A triangle that has one interior angle that measures more than 90° is an obtuse triangle or obtuse-angled triangle.

A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the same measure, all three sides have the same length, and such a triangle is therefore equilateral.

<#"justify" height="78" src="doc_zip25.jpg" /> <#"justify" height="67" src="doc_zip26.jpg" /> <#"justify">RightObtuseAcute

. Quadrilateral

A quadrilateral is a polygon <#"justify">There are three topological types of quadrilaterals convex quadrilaterals (left figure), concave quadrilaterals (middle figure), and crossed quadrilaterals (or butterflies, or bow-ties; right figure).

In addition to this there are many types of convex quadrilaterals, such as a parallelogram, rectangle, rhombus, square, trapezoid <#"99" src="doc_zip28.jpg" />

A rectangle <#"87" src="doc_zip29.jpg" />

rhombus <#"justify">

A square <#"108" src="doc_zip31.jpg" />

A trapezoid <#"84" src="doc_zip32.jpg" />

kite has two pairs of sides. Each pair is made up of adjacent sides that are equal in length. The angles are equal where the pairs meet. Diagonals (dashed lines) meet at a right angle, and one of the diagonal bisects (cuts equally in half) the other.

4. Circles

circle is a simple shape <#"8" height="41" src="doc_zip34.jpg" /> from the center <#"41" src="doc_zip35.jpg" /> is called the center <#"41" src="doc_zip36.jpg" />. The angle a circle subtends from its center is a full angle <#"41" src="doc_zip37.jpg" /> or radians <#"160" src="doc_zip39.jpg" />

are two main "slices" of a circle"pizza" slice is called a sector <#"134" src="doc_zip40.jpg" />


this paper I have tried to introduce several two-dimensional geometric figures and some of their properties. I introduced triangles, quadrilaterals, polygons, their classifying by different bases, circle and its parts, which are used for solving practical the course of geometry consist of two parts - plan geometry and space geometry. In school courses plane geometry or two-dimensional geometric figures are studied in 7th, 8th and 9th grade and they are the basis of geometry. Space geometry or three-dimensional geometric shapes are studied in 10th, 11th and 12th grade, but during the study of three-dimensional geometric shapes two-dimensional geometric figures are widely used. In addition to this two-dimensional geometric figures are defined as parts of three-dimensional and n-dimensional shapes. Our world is three- dimensional, that's why we must know common properties of three-dimensional geometric shapes, therefore we must recognize also two-dimensional geometric figures.dimensional geometric figures are widely used in the practise, so everybody must know some properties of geometric figures and their applications.geometry is very beautiful and practice subject and it develops the students analytical and visual thinking, which explains its essential role in our life.


1.Point- points are zero-dimensional <#"41" src="doc_zip41.jpg" />-gon- A polygon with vertices and sides is known as an -gon

7.Convex <#"justify">18.Acute triangle or acute-angled triangle.- A triangle that has all interior angles measuring less than 90°

19.Obtuse triangle or obtuse-angled triangle- A triangle that has one interior angle that measures more than 90°

20.Quadrilateral - A quadrilateral is a polygon <#"justify">References

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Armenian state pedagogical university after khachatur abovyan Faculty of Mathematics, Physics and Informatics

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