Geometric solid figures are solid bodies that occupy a non-zero volume in Euclidean (three-dimensional) space. These figures are studied by a branch of mathematics called “spatial geometry”. Knowledge about the properties of three-dimensional figures is used in engineering and the natural sciences. In the article we will consider the question of geometric three-dimensional figures and their names.
Geometric solids
Since these bodies have a finite dimension in three spatial directions, a system of three coordinate axes is used to describe them in geometry. These axes have the following properties:
- They are orthogonal to each other, that is, perpendicular.
- These axes are normalized, meaning the basis vectors of each axis are the same length.
- Any of the coordinate axes is the result of the vector product of the other two.
Speaking about geometric volumetric figures and their names, it should be noted that they all belong to one of 2 large classes:
- Class of polyhedra. These figures, based on the name of the class, have straight edges and flat faces. A face is a plane that limits a shape. The point where two faces join is called an edge, and the point where three faces join is called a vertex. Polyhedra include the geometric figure of a cube, tetrahedrons, prisms, and pyramids. For these figures, Euler's theorem is valid, which establishes a connection between the number of sides (C), edges (P) and vertices (B) for each polyhedron. Mathematically, this theorem is written as follows: C + B = P + 2.
- Class of round bodies or bodies of revolution. These figures have at least one surface forming them that is curved. For example, a ball, a cone, a cylinder, a torus.
As for the properties of volumetric figures, the two most important of them should be highlighted:
- The presence of a certain volume that a figure occupies in space.
- The presence of a surface area for each volumetric figure.
Both properties for each figure are described by specific mathematical formulas.
Let us consider below the simplest geometric volumetric figures and their names: cube, pyramid, prism, tetrahedron and ball.
Cube figure: description
The geometric figure cube is a three-dimensional body formed by 6 square planes or surfaces. This figure is also called a regular hexahedron, since it has 6 sides, or a rectangular parallelepiped, since it consists of 3 pairs of parallel sides that are mutually perpendicular to each other. It is called a cube whose base is a square and whose height is equal to the side of the base.
Since a cube is a polyhedron or polyhedron, Euler's theorem can be applied to it to determine the number of its edges. Knowing that the number of sides is 6, and the cube has 8 vertices, the number of edges is: P = C + B - 2 = 6 + 8 - 2 = 12.
If we denote the length of the side of a cube by the letter “a”, then the formulas for its volume and surface area will look like: V = a 3 and S = 6*a 2, respectively.
Pyramid figure
A pyramid is a polyhedron that consists of a simple polyhedron (the base of the pyramid) and triangles that connect to the base and have one common vertex (the top of the pyramid). The triangles are called the lateral faces of the pyramid.
The geometric characteristics of a pyramid depend on which polygon lies at its base, as well as on whether the pyramid is straight or oblique. A straight pyramid is understood to be a pyramid for which a straight line perpendicular to the base, drawn through the top of the pyramid, intersects the base at its geometric center.
One of the simple pyramids is a quadrangular straight pyramid, at the base of which lies a square with side “a”, the height of this pyramid is “h”. For this pyramid figure, the volume and surface area will be equal: V = a 2 *h/3 and S = 2*a*√(h 2 +a 2 /4) + a 2, respectively. Applying Euler's theorem for it, taking into account that the number of faces is 5 and the number of vertices is 5, we obtain the number of edges: P = 5 + 5 - 2 = 8.
Tetrahedron figure: description
The geometric figure tetrahedron is understood as a three-dimensional body formed by 4 faces. Based on the properties of space, such faces can only represent triangles. Thus, a tetrahedron is a special case of a pyramid, which has a triangle at its base.
If all 4 triangles forming the faces of a tetrahedron are equilateral and equal to each other, then such a tetrahedron is called regular. This tetrahedron has 4 faces and 4 vertices, the number of edges is 4 + 4 - 2 = 6. Applying standard formulas from plane geometry for the figure in question, we obtain: V = a 3 * √2/12 and S = √3*a 2, where a is the length of the side of an equilateral triangle.
It is interesting to note that in nature some molecules have the shape of a regular tetrahedron. For example, a methane molecule CH 4, in which the hydrogen atoms are located at the vertices of the tetrahedron and are connected to the carbon atom by covalent chemical bonds. The carbon atom is located at the geometric center of the tetrahedron.
The tetrahedron shape, which is easy to manufacture, is also used in engineering. For example, the tetrahedral shape is used in the manufacture of anchors for ships. Note that NASA's Mars Pathfinder space probe, which landed on the surface of Mars on July 4, 1997, also had the shape of a tetrahedron.
Prism figure
This geometric figure can be obtained by taking two polyhedra, placing them parallel to each other in different planes of space, and connecting their vertices accordingly. The result will be a prism, two polyhedra are called its bases, and the surfaces connecting these polyhedra will have the shape of parallelograms. A prism is called straight if its sides (parallelograms) are rectangles.
A prism is a polyhedron, therefore it is true for it. For example, if the base of the prism is a hexagon, then the number of sides of the prism is 8, and the number of vertices is 12. The number of edges will be equal to: P = 8 + 12 - 2 = 18. For a straight line a prism of height h, at the base of which lies a regular hexagon with side a, the volume is equal to: V = a 2 *h*√3/4, the surface area is equal to: S = 3*a*(a*√3 + 2*h).
Speaking about simple geometric volumetric figures and their names, we should mention the ball. A volumetric body called a ball is understood as a body that is limited to a sphere. In turn, a sphere is a collection of points in space equidistant from one point, which is called the center of the sphere.
Since the ball belongs to the class of round bodies, there is no concept of sides, edges and vertices for it. the sphere bounding the ball is found by the formula: S = 4*pi*r 2, and the volume of the ball can be calculated by the formula: V = 4*pi*r 3 /3, where pi is the number pi (3.14), r - radius of the sphere (ball).
Lesson Objectives:
- Cognitive: create conditions for familiarization with concepts flat And volumetric geometric shapes, expand your understanding of the types of volumetric figures, teach how to determine the type of figure, and compare figures.
- Communicative: create conditions for developing the ability to work in pairs and groups; fostering a friendly attitude towards each other; to cultivate mutual assistance and mutual assistance among students.
- Regulatory: create conditions for the formation to plan an educational task, build a sequence of necessary operations, adjust your activities.
- Personal: create conditions for the development of computing skills, logical thinking, interest in mathematics, the formation of cognitive interests, intellectual abilities of students, independence in acquiring new knowledge and practical skills.
Planned results:
personal:
- formation of cognitive interests and intellectual abilities of students; formation of value relations towards each other;
independence in acquiring new knowledge and practical skills; - formation of skills to perceive, process received information, and highlight the main content.
meta-subject:
- mastering the skills of independent acquisition of new knowledge;
- organization of educational activities, planning;
- development of theoretical thinking based on the formation of skills to establish facts.
subject:
- master the concepts of flat and three-dimensional figures, learn to compare figures, find flat and three-dimensional figures in the surrounding reality, learn to work with development.
UUD general scientific:
- search and selection of necessary information;
- application of information retrieval methods, conscious and arbitrary construction of speech utterances orally.
UUD personal:
- evaluate your own and others’ actions;
- demonstration of trust, attentiveness, goodwill;
- ability to work in pairs;
- express a positive attitude towards the learning process.
Equipment: textbook, interactive whiteboard, emoticons, models of figures, development of figures, individual traffic lights, rectangles - means of feedback, Explanatory dictionary.
Lesson type: learning new material.
Methods: verbal, research, visual, practical.
Forms of work: frontal, group, pair, individual.
1. Organization of the beginning of the lesson.
In the morning the sun rose.
A new day has been brought to us.
Strong and kind
We are celebrating a new day.
Here are my hands, I open them
Them towards the sun.
Here are my legs, they are firm
They stand on the ground and lead
Me on the right path.
Here is my soul, I reveal
Her towards people.
Come, new day!
Hello new day!
2. Updating knowledge.
Let's create good mood. Smile at me and at each other, sit down!
To reach your goal, you must first go.
There is a statement in front of you, read it. What does this statement mean?
(To achieve something, you need to do something)
And indeed, guys, only those who prepare themselves to be collected and organized in their actions can hit the target. And so I hope that you and I will achieve our goal in this lesson.
Let's begin our journey to achieving the goal of today's lesson.
3. Preparatory work.
Look at the screen. What do you see? (Geometric figures)
Name these figures.
What task can you offer to your classmates? (divide the shapes into groups)
You have cards with these figures on your desks. Complete this task in pairs.
On what basis did you divide these figures?
- Flat and volumetric figures
- Based on volumetric figures
What figures have we already worked with? What did you learn to find from them? What figures do we encounter for the first time in geometry?
What is the topic of our lesson? (The teacher adds words on the board: volumetric, the lesson topic appears on the board: Volumetric geometric shapes.)
What should we learn in class?
4. “Discovery” of new knowledge in practical research work.
(The teacher shows a cube and a square.)
How are they similar?
Can we say that these are the same thing?
What is the difference between a cube and a square?
Let's do an experiment. (Students receive individual figures - cube and square.)
Let's try to attach the square to the flat surface of the port. What do we see? Did he lay down (entirely) on the surface of the desk? Close?
! What do we call a figure that can be placed entirely on one flat surface? (Flat figure.)
Is it possible to press the cube completely (entirely) to the desk? Let's check.
Can a cube be called a flat figure? Why? Is there space between your hand and the desk?
! So what can we say about the cube? (Occupies a certain space, is a three-dimensional figure.)
CONCLUSIONS: What is the difference between flat and three-dimensional figures? (The teacher posts conclusions on the board.)
- Can be placed entirely on one flat surface.
VOLUMETRIC
- occupy a certain space,
- rise above a flat surface.
Volumetric figures: pyramid, cube, cylinder, cone, ball, parallelepiped.
4. Discovery of new knowledge.
1. Name the figures shown in the picture.
What shape do the bases of these figures have?
What other shapes can be seen on the surface of a cube and a prism?
2. Figures and lines on the surface of volumetric figures have their own names.
Suggest your names.
The sides that form a flat figure are called faces. And the lateral lines are the ribs. The corners of polygons are vertices. These are elements of volumetric figures.
Guys, what do you think, what are the names of such three-dimensional figures that have many sides? Polyhedra.
Working with notebooks: reading new material
Correlation between real objects and volumetric bodies.
Now select for each object the three-dimensional figure that it resembles.
The box is a parallelepiped.
- An apple is a ball.
- Pyramid - pyramid.
- The jar is a cylinder.
- Flower pot - cone.
- The cap is a cone.
- The vase is a cylinder.
- The ball is a ball.
5. Physical exercise.
1. Imagine a big ball, stroke it from all sides. It's big and smooth.
(Students “wrap” their hands around and stroke an imaginary ball.)
Now imagine a cone, touch its top. The cone grows upward, now it is already taller than you. Jump to the top of it.
Imagine that you are inside a cylinder, pat its upper base, stomp on the lower one, and now with your hands along the side surface.
The cylinder became a small gift box. Imagine that you are a surprise that is in this box. I press the button and... a surprise pops out of the box!
6. Group work:
(Each group receives one of the figures: a cube, a pyramid, a parallelepiped. The children study the resulting figure, and write down the conclusions on a card prepared by the teacher.)
Group 1.(To study the parallelepiped)
Group 2.(For studying the pyramid)
Group 3.(For studying the cube)
7. Crossword solution
8. Lesson summary. Reflection of activity.
Crossword solution in presentation
What new things have you discovered for yourself today?
All geometric shapes can be divided into three-dimensional and flat.
And I learned the names of three-dimensional figures
Lesson Objectives:
- Cognitive: create conditions for familiarization with concepts flat And volumetric geometric shapes, expand your understanding of the types of volumetric figures, teach how to determine the type of figure, and compare figures.
- Communicative: create conditions for developing the ability to work in pairs and groups; fostering a friendly attitude towards each other; to cultivate mutual assistance and mutual assistance among students.
- Regulatory: create conditions for the formation to plan an educational task, build a sequence of necessary operations, adjust your activities.
- Personal: create conditions for the development of computing skills, logical thinking, interest in mathematics, the formation of cognitive interests, intellectual abilities of students, independence in acquiring new knowledge and practical skills.
Planned results:
personal:
- formation of cognitive interests and intellectual abilities of students; formation of value relations towards each other;
independence in acquiring new knowledge and practical skills; - formation of skills to perceive, process received information, and highlight the main content.
meta-subject:
- mastering the skills of independent acquisition of new knowledge;
- organization of educational activities, planning;
- development of theoretical thinking based on the formation of skills to establish facts.
subject:
- master the concepts of flat and three-dimensional figures, learn to compare figures, find flat and three-dimensional figures in the surrounding reality, learn to work with development.
UUD general scientific:
- search and selection of necessary information;
- application of information retrieval methods, conscious and arbitrary construction of speech utterances orally.
UUD personal:
- evaluate your own and others’ actions;
- demonstration of trust, attentiveness, goodwill;
- ability to work in pairs;
- express a positive attitude towards the learning process.
Equipment: textbook, interactive whiteboard, emoticons, models of figures, development of figures, individual traffic lights, rectangles - means of feedback, Explanatory dictionary.
Lesson type: learning new material.
Methods: verbal, research, visual, practical.
Forms of work: frontal, group, pair, individual.
1. Organization of the beginning of the lesson.
In the morning the sun rose.
A new day has been brought to us.
Strong and kind
We are celebrating a new day.
Here are my hands, I open them
Them towards the sun.
Here are my legs, they are firm
They stand on the ground and lead
Me on the right path.
Here is my soul, I reveal
Her towards people.
Come, new day!
Hello new day!
2. Updating knowledge.
Let's create a good mood. Smile at me and at each other, sit down!
To reach your goal, you must first go.
There is a statement in front of you, read it. What does this statement mean?
(To achieve something, you need to do something)
And indeed, guys, only those who prepare themselves to be collected and organized in their actions can hit the target. And so I hope that you and I will achieve our goal in this lesson.
Let's begin our journey to achieving the goal of today's lesson.
3. Preparatory work.
Look at the screen. What do you see? (Geometric figures)
Name these figures.
What task can you offer to your classmates? (divide the shapes into groups)
You have cards with these figures on your desks. Complete this task in pairs.
On what basis did you divide these figures?
- Flat and volumetric figures
- Based on volumetric figures
What figures have we already worked with? What did you learn to find from them? What figures do we encounter for the first time in geometry?
What is the topic of our lesson? (The teacher adds words on the board: volumetric, the lesson topic appears on the board: Volumetric geometric shapes.)
What should we learn in class?
4. “Discovery” of new knowledge in practical research work.
(The teacher shows a cube and a square.)
How are they similar?
Can we say that these are the same thing?
What is the difference between a cube and a square?
Let's do an experiment. (Students receive individual figures - cube and square.)
Let's try to attach the square to the flat surface of the port. What do we see? Did he lay down (entirely) on the surface of the desk? Close?
! What do we call a figure that can be placed entirely on one flat surface? (Flat figure.)
Is it possible to press the cube completely (entirely) to the desk? Let's check.
Can a cube be called a flat figure? Why? Is there space between your hand and the desk?
! So what can we say about the cube? (Occupies a certain space, is a three-dimensional figure.)
CONCLUSIONS: What is the difference between flat and three-dimensional figures? (The teacher posts conclusions on the board.)
- Can be placed entirely on one flat surface.
VOLUMETRIC
- occupy a certain space,
- rise above a flat surface.
Volumetric figures: pyramid, cube, cylinder, cone, ball, parallelepiped.
4. Discovery of new knowledge.
1. Name the figures shown in the picture.
What shape do the bases of these figures have?
What other shapes can be seen on the surface of a cube and a prism?
2. Figures and lines on the surface of volumetric figures have their own names.
Suggest your names.
The sides that form a flat figure are called faces. And the lateral lines are the ribs. The corners of polygons are vertices. These are elements of volumetric figures.
Guys, what do you think, what are the names of such three-dimensional figures that have many sides? Polyhedra.
Working with notebooks: reading new material
Correlation between real objects and volumetric bodies.
Now select for each object the three-dimensional figure that it resembles.
The box is a parallelepiped.
- An apple is a ball.
- Pyramid - pyramid.
- The jar is a cylinder.
- Flower pot - cone.
- The cap is a cone.
- The vase is a cylinder.
- The ball is a ball.
5. Physical exercise.
1. Imagine a big ball, stroke it from all sides. It's big and smooth.
(Students “wrap” their hands around and stroke an imaginary ball.)
Now imagine a cone, touch its top. The cone grows upward, now it is already taller than you. Jump to the top of it.
Imagine that you are inside a cylinder, pat its upper base, stomp on the lower one, and now with your hands along the side surface.
The cylinder became a small gift box. Imagine that you are a surprise that is in this box. I press the button and... a surprise pops out of the box!
6. Group work:
(Each group receives one of the figures: a cube, a pyramid, a parallelepiped. The children study the resulting figure, and write down the conclusions on a card prepared by the teacher.)
Group 1.(To study the parallelepiped)
Group 2.(For studying the pyramid)
Group 3.(For studying the cube)
7. Crossword solution
8. Lesson summary. Reflection of activity.
Crossword solution in presentation
What new things have you discovered for yourself today?
All geometric shapes can be divided into three-dimensional and flat.
And I learned the names of three-dimensional figures
Geometric figures are closed sets of points on a plane or in space that are limited by a finite number of lines. They can be linear (1D), planar (2D) or spatial (3D).
Any body that has a shape is a collection of geometric shapes.
Any figure can be described by a mathematical formula of varying degrees of complexity. Starting from a simple mathematical expression to the sum of a series of mathematical expressions.
The main mathematical parameters of geometric figures are radii, lengths of sides or edges and angles between them.
Below are the basic geometric figures most often used in applied calculations, formulas and links to calculation programs.
Linear geometric shapes
1. PointA point is the basic measurement object. The main and only mathematical characteristic of a point is its coordinate.
2. Line
A line is a thin spatial object that has a finite length and is a chain of points connected to each other. The main mathematical characteristic of a line is its length.
A ray is a thin spatial object of infinite length and representing a chain of points connected to each other. The main mathematical characteristics of the ray are the coordinate of its origin and direction.
Flat geometric shapes
1. CircleA circle is a geometric locus of points on a plane, the distance from which to its center does not exceed a given number, called the radius of this circle. The main mathematical characteristic of a circle is its radius.
2. Square
A square is a quadrilateral in which all angles and all sides are equal. The main mathematical characteristic of a square is the length of its side.
3. Rectangle
A rectangle is a quadrilateral whose angles are all 90 degrees (right). The main mathematical characteristics of a rectangle are the lengths of its sides.
4. Triangle
A triangle is a geometric figure formed by three segments that connect three points (vertices of the triangle) that do not lie on the same straight line. The main mathematical characteristics of a triangle are the lengths of the sides and height.
5. Trapezoid
A trapezoid is a quadrilateral in which two sides are parallel and the other two sides are not parallel. The main mathematical characteristics of a trapezoid are the lengths of the sides and height.
6. Parallelogram
A parallelogram is a quadrilateral whose opposite sides are parallel. The main mathematical characteristics of a parallelogram are the lengths of its sides and height. 
A rhombus is a quadrilateral that has all sides, but the angles of its vertices are not equal to 90 degrees. The main mathematical characteristics of a rhombus are the length of its side and its height.
8. Ellipse
An ellipse is a closed curve on a plane, which can be represented as an orthogonal projection of a section of the circumference of a cylinder onto a plane. The main mathematical characteristics of a circle are the length of its semi-axes.
Volumetric geometric shapes
1. BallA ball is a geometric body, which is a collection of all points in space located at a given distance from its center. The main mathematical characteristic of a ball is its radius.
A sphere is the shell of a geometric body, which is a collection of all points in space located at a given distance from its center. The main mathematical characteristic of a sphere is its radius.
A cube is a geometric body that is a regular polyhedron, each face of which is a square. The main mathematical characteristic of a cube is the length of its edge.
4. Parallelepiped
A parallelepiped is a geometric body, which is a polyhedron with six faces and each of them is a rectangle. The main mathematical characteristics of a parallelepiped are the lengths of its edges.
5. Prism
A prism is a polyhedron, two of whose faces are equal polygons lying in parallel planes, and the remaining faces are parallelograms having common sides with these polygons. The main mathematical characteristics of a prism are the base area and height.
A cone is a geometric figure obtained by combining all the rays emanating from one vertex of the cone and passing through a flat surface. The main mathematical characteristics of a cone are the radius of the base and the height.
7. Pyramid
A pyramid is a polyhedron whose base is an arbitrary polygon, and the side faces are triangles that have a common vertex. The main mathematical characteristics of a pyramid are the base area and height.
8. Cylinder
A cylinder is a geometric figure bounded by a cylindrical surface and two parallel planes intersecting it. The main mathematical characteristics of a cylinder are the base radius and height.
You can quickly perform these simple mathematical operations using our online programs. To do this, enter the initial value in the appropriate field and click the button.
This page presents all the geometric figures that are most often found in geometry to represent an object or part of it on a plane or in space.
There are an infinite number of forms. Shape is the external outline of an object.
The study of shapes can begin from early childhood, drawing your child’s attention to the world around us, which consists of shapes (a plate is round, a TV is rectangular).
From the age of two, a child should know three simple shapes - a circle, a square, a triangle. At first he should just show them when you ask. And at three years old, you can already name them yourself and distinguish a circle from an oval, a square from a rectangle.
The more exercises a child does to consolidate shapes, the more new shapes he will remember.
The future first-grader must know all the simple geometric shapes and be able to make applications from them.
What do we call a geometric figure?
A geometric figure is a standard with which you can determine the shape of an object or its parts.
Figures are divided into two groups: flat figures, three-dimensional figures.
We call plane figures those figures that are located in the same plane. These include circle, oval, triangle, quadrangle (rectangle, square, trapezoid, rhombus, parallelogram) and all kinds of polygons.
Three-dimensional figures include: sphere, cube, cylinder, cone, pyramid. These are those shapes that have height, width and depth.
Follow two simple tips when explaining geometric figures:
- Patience. What seems simple and logical to us, adults, will seem simply incomprehensible to a child.
- Try drawing shapes with your child.
- A game. Start learning shapes in a playful way. Good exercises for consolidating and studying flat shapes are applications from geometric shapes. For voluminous ones, you can use ready-made store-bought games, and also choose applications where you can cut out and glue a voluminous shape.