All figures and their names. "Volume geometric shapes

Figure is an arbitrary set of points on the plane. A point, a straight line, a segment, a ray, a triangle, a circle, a square, and so on are all examples of geometric shapes.

Dot- the basic concept of geometry, it is an abstract object that has no measurement characteristics: no height, no length, no radius.

Line- this is a set of points sequentially located one after another. Only the length of a line is measured. It has no width or thickness.

Straight line- this is a line that does not bend, has neither beginning nor end, it can be continued endlessly in both directions.

Beam- this is part of a straight line that has a beginning but no end; it can be continued endlessly in only one direction.

Segment is a part of a straight line bounded by two points. A line segment has a beginning and an end, so its length can be measured.

Crooked line is a smoothly curving line, which is determined by the location of its constituent points.

broken line is a figure that consists of segments connected in series at their ends.

Vertices of a broken line- This

1. the point from which the broken line begins,
2. points at which the segments that form a broken line are connected,
3. the point at which the broken line ends.

Links of a broken line– these are the segments that make up the broken line. The number of links of a polyline is always 1 less than the number of vertices of a polyline.

Open line is a line whose ends are not connected together.

Closed line is a line whose ends are connected together.

Polygon is a closed broken line. The vertices of the polygon are called the vertices of the polygon, and the segments are called the sides of the polygon.

In this lesson you will learn what it is geometric shapes. We will talk about figures depicted on a plane and their properties. You will learn about the simplest forms of geometric shapes such as dots and lines. Consider how a segment and a ray are formed. Get to know the definition and various types corners The next shape whose definition and properties are discussed in this lesson is a circle. The following discusses the definition of triangle and polygon and their varieties.

Rice. 10. Circle and circumference

Think about which points belong to a circle and which circles (see Fig. 11).

Rice. 11. Mutual position dots and circle, dots and circle

Correct answer: points and belong to the circle, and only points and belong to the circle.

A point is the center of a circle or circle. Segments are the radii of a circle or circle, that is, segments that connect the center and any point lying on the circle. A segment is the diameter of a circle or circle, that is, it is a segment connecting two points lying on the circle and passing through the center. The radius is half the diameter (see Fig. 12).

Rice. 12. Radius and diameter

Let's now remember what kind of figure is called a triangle. A triangle is a geometric figure consisting of three points that do not lie on the same straight line and three segments connecting these points in pairs. A triangle has three angles.

Consider a triangle (see Fig. 13).

Rice. 13. Triangle

It has three angles - corner, corner and corner. The points , , are called the vertices of the triangle. Three segments - segment , , - are the sides of the triangle.

Let us repeat what types of triangles are distinguished (see Fig. 14).

Rice. 14. Types of triangles

Based on the types of angles, triangles can be divided into acute, rectangular and obtuse. In a triangle, all angles are acute; such a triangle is called acute. A triangle has a right angle, such a triangle is called a right triangle. A triangle has an obtuse angle, such a rectangle is called an obtuse triangle.

Triangles are distinguished based on whether the lengths of the sides are equal:

Scalene - such triangles have different lengths of all sides;

Equilateral - these triangles have equal lengths of all sides;

Isosceles - their two sides have the same lengths. Two sides of equal length are called the lateral sides of the triangle, and the third side is the base of the triangle (see Fig. 15).

Rice. 15. Types of triangles

What shapes are called polygons? If you sequentially connect several points so that their connection gives a closed broken line, then an image of a polygon, quadrangle, pentagon or hexagon, etc. is created.

Polygons are named by the number of angles. Each polygon has as many vertices and sides as there are angles (see Fig. 16).

Rice. 16. Polygons

All the figures depicted (see Fig. 17) are called quadrilaterals. Why?

Rice. 17. Quadrilaterals

You probably noticed that all the figures have four corners, but they can all be divided into two groups. How would you do it?

You probably separated quadrilaterals in which all angles are right angles into a separate group, and such quadrilaterals were called rectangular quadrilaterals. The opposite sides of the rectangles are equal (see Fig. 18).

Rice. 18. Rectangular quadrilaterals

In a rectangle and - opposite sides, and they are equal, and are also opposite sides, and they are equal (see Fig. 19).

Lesson topic

Geometric shapes

What is a geometric figure

Geometric figures are a collection of many points, lines, surfaces or bodies that are located on a surface, plane or space and form a finite number of lines.

The term “figure” is to some extent formally applied to a set of points, but as a rule, a figure is usually called a set that is located on a plane and is limited by a finite number of lines.

A point and a straight line are the basic geometric figures located on a plane.

The simplest geometric figures on a plane include a segment, a ray and a broken line.

What is geometry

Geometry is like this mathematical science, which studies the properties of geometric shapes. If we literally translate the term “geometry” into Russian, it means “land surveying,” since in ancient times the main task of geometry as a science was the measurement of distances and areas on the surface of the earth.

The practical application of geometry is invaluable at all times and regardless of profession. Neither a worker, nor an engineer, nor an architect, nor even an artist can do without knowledge of geometry.

In geometry there is a section that deals with the study of various figures on a plane and is called planimetry.

You already know that a figure is an arbitrary set of points located on a plane.

Geometric figures include: point, straight line, segment, ray, triangle, square, circle and other figures that are studied by planimetry.

Dot

From the material studied above, you already know that the point refers to the main geometric figures. And although this is the smallest geometric figure, it is necessary for constructing other figures on a plane, drawing or image and is the basis for all other constructions. After all, the construction of more complex geometric figures consists of many points characteristic of a given figure.

In geometry, points represent in capital letters Latin alphabet, for example, such as: A, B, C, D....

Now let’s summarize, and so, from a mathematical point of view, a point is such an abstract object in space that does not have volume, area, length and other characteristics, but remains one of the fundamental concepts in mathematics. A point is a zero-dimensional object that has no definition. According to Euclid's definition, a point is something that cannot be defined.

Straight

Like a point, a straight line refers to figures on a plane, which has no definition, since it consists of an infinite number of points located on one line, which has neither beginning nor end. It can be argued that a straight line is infinite and has no limit.

If a straight line begins and ends with a point, then it is no longer a straight line and is called a segment.

But sometimes a straight line has a point on one side and not on the other. In this case, the straight line turns into a beam.

If you take a straight line and put a point in its middle, then it will split the straight line into two oppositely directed rays. These rays are additional.

If in front of you there are several segments connected to each other so that the end of the first segment becomes the beginning of the second, and the end of the second segment becomes the beginning of the third, etc., and these segments are not on the same straight line and when connected have a common point, then such the chain is a broken line.

Exercise

Which broken line is called unclosed?
How is a straight line designated?
What is the name of a broken line that has four closed links?
What is the name of a broken line with three closed links?

When the end of the last segment of a broken line coincides with the beginning of the 1st segment, then such a broken line is called closed. An example of a closed polyline is any polygon.

Plane

Like a point and a straight line, a plane is a primary concept; it has no definition and one cannot see either a beginning or an end. Therefore, when considering a plane, we consider only that part of it that is limited by a closed broken line. Thus, any smooth surface can be considered a plane. This surface can be a sheet of paper or a table.

Corner

A figure that has two rays and a vertex is called an angle. The junction of the rays is the vertex of this angle, and its sides are the rays that form this angle.

Exercise:

1. How is an angle indicated in the text?
2. What units can you use to measure an angle?
3. What are the angles?

Parallelogram

A parallelogram is a quadrilateral whose opposite sides are parallel in pairs.

Rectangle, square and rhombus are special cases of parallelogram.

A parallelogram with right angles equal to 90 degrees is a rectangle.

A square is the same parallelogram; its angles and sides are equal.

As for the definition of a rhombus, it is a geometric figure whose all sides are equal.

In addition, you should know that every square is a rhombus, but not every rhombus can be a square.

Trapezoid

When considering a geometric figure such as a trapezoid, we can say that, in particular, like a quadrilateral, it has one pair of parallel opposite sides and is curvilinear.

Circle and Circle

A circle is the geometric locus of points on a plane equidistant from a given point, called the center, at a given non-zero distance, called its radius.

Triangle

The triangle you have already studied also belongs to simple geometric figures. This is one of the types of polygons in which part of the plane is limited by three points and three segments that connect these points in pairs. Any triangle has three vertices and three sides.

Exercise: Which triangle is called degenerate?

Polygon

Polygons include geometric shapes different forms, which have a closed broken line.

In a polygon, all points that connect the segments are its vertices. And the segments that make up a polygon are its sides.

Did you know that the emergence of geometry goes back centuries and is associated with the development of various crafts, culture, art and observation of the surrounding world. And the name of geometric figures is confirmation of this, since their terms did not arise just like that, but due to their similarity and similarity.

After all, the term “trapezoid” translated from the ancient Greek language from the word “trapezion” means table, meal and other derivative words.

“Cone” comes from the Greek word “konos,” which means pine cone.

“Line” has Latin roots and comes from the word “linum”, translated it sounds like linen thread.

Did you know that if you take geometric shapes with the same perimeter, then among them the circle turns out to have the largest area.

Geometric figures are a complex of points, lines, solids or surfaces. These elements can be located both on the plane and in space, forming a finite number of straight lines.

The term “figure” implies several sets of points. They must be located on one or more planes and at the same time limited to a specific number of completed lines.

The main geometric figures are the point and the straight line. They are located on a plane. In addition to them, among simple figures there is a ray, a broken line and a segment.

Dot

This is one of the main figures of geometry. It is very small, but it is always used for building various forms on a plane. The point is the main figure for absolutely all constructions, even the highest complexity. In geometry, it is usually denoted by a letter of the Latin alphabet, for example, A, B, K, L.

From a mathematical point of view, a point is an abstract spatial object that does not have such characteristics as area or volume, but at the same time remains a fundamental concept in geometry. This zero-dimensional object simply has no definition.

Straight

This figure is completely placed in one plane. A straight line does not have a specific mathematical definition, since it consists of a huge number of points located on one infinite line, which has no limit or boundaries.

There is also a segment. This is also a straight line, but it starts and ends from a point, which means it has geometric restrictions.

The line can also turn into a directional beam. This happens when a straight line starts from a point, but does not have a clear ending. If you put a point in the middle of the line, then it will split into two rays (additional), and oppositely directed to each other.

Several segments that are sequentially connected to each other by ends in common point and are not located on the same straight line, it is usually called a broken line.

Corner

Geometric figures, the names of which we discussed above, are considered key elements used in the construction of more complex models.

An angle is a structure consisting of a vertex and two rays that extend from it. That is, the sides of this figure connect at one point.

Plane

Let's consider another primary concept. A plane is a figure that has neither end nor beginning, as well as a straight line and a point. When considering this geometric element, only its part, limited by the contours of a broken closed line, is taken into account.

Any smooth bounded surface can be considered a plane. This could be an ironing board, a piece of paper, or even a door.

Quadrilaterals

A parallelogram is a geometric figure whose opposite sides are parallel to each other in pairs. Among the particular types of this design are diamond, rectangle and square.

A rectangle is a parallelogram in which all sides touch at right angles.

A square is a quadrilateral with equal sides and angles.

A rhombus is a figure in which all sides are equal. In this case, the angles can be completely different, but in pairs. Each square is considered a diamond. But in the opposite direction this rule does not always apply. Not every rhombus is a square.

Trapezoid

Geometric shapes can be completely different and bizarre. Each of them has a unique shape and properties.

A trapezoid is a figure that is somewhat similar to a quadrilateral. It has two parallel opposite sides and is considered curved.

Circle

This geometric figure implies the location on one plane of points equidistant from its center. In this case, a given non-zero segment is usually called a radius.

Triangle

This is a simple geometric figure that is very often encountered and studied.

A triangle is considered a subtype of a polygon, located on one plane and limited by three edges and three points of contact. These elements are connected in pairs.

Polygon

The vertices of polygons are the points connecting the segments. And the latter, in turn, are considered to be parties.

Volumetric geometric shapes

• prism;
• sphere;
• cone;
• cylinder;
• pyramid;

These bodies have something in common. All of them are limited to a closed surface, inside of which there are many points.

Volumetric bodies are studied not only in geometry, but also in crystallography.

Curious facts

Surely you will be interested in reading the information provided below.

• Geometry was formed as a science back in ancient times. This phenomenon is usually associated with the development of art and various crafts. And the names of geometric figures indicate the use of the principles of determining similarity and similarity.
• Translated from ancient Greek, the term “trapezoid” means a table for a meal.
• If you take different shapes whose perimeter is the same, then the circle is guaranteed to have the largest area.
• Translated from Greek, the term “cone” means a pine cone.
• There is a famous painting by Kazemir Malevich, which, since the last century, has attracted the views of many painters. The work “Black Square” has always been mystical and mysterious. The geometric figure on the white canvas delights and amazes at the same time.

Exists large number geometric shapes. They all differ in parameters, and sometimes even surprise in shape.