A body whose surface consists of a finite. A polyhedron is a body whose surface consists of a finite number of flat polygons. Polyhedron. It turns out that calcite crystals, no matter how much they are crushed into smaller parts, always disintegrate
Cube, ball, pyramid, cylinder, cone - geometric bodies. Among them are polyhedra. Polyhedron is a geometric body whose surface consists of a finite number of polygons. Each of these polygons is called a face of the polyhedron, the sides and vertices of these polygons are, respectively, the edges and vertices of the polyhedron.
Dihedral angles between adjacent faces, i.e. faces that have a common side - the edge of the polyhedron - are also dihedral minds of the polyhedron. The angles of polygons - the faces of a convex polygon - are flat minds of the polyhedron. In addition to flat and dihedral angles, a convex polyhedron also has polyhedral angles. These angles form faces that have a common vertex.
Among the polyhedra there are prisms And pyramids.
Prism - is a polyhedron whose surface consists of two equal polygons and parallelograms that have common sides with each of the bases.
Two equal polygons are called reasons ggrizmg, and parallelograms are her lateral edges. The side faces form lateral surface prisms. Edges that do not lie at the base are called lateral ribs prisms.
The prism is called p-coal, if its bases are i-gons. In Fig. 24.6 shows a quadrangular prism ABCDA"B"C"D".
The prism is called straight, if its side faces are rectangles (Fig. 24.7).
The prism is called correct , if it is straight and its bases are regular polygons.
A quadrangular prism is called parallelepiped , if its bases are parallelograms.
The parallelepiped is called rectangular, if all its faces are rectangles.
Diagonal of a parallelepiped is a segment connecting its opposite vertices. A parallelepiped has four diagonals.
It has been proven that The diagonals of a parallelepiped intersect at one point and are bisected by this point. The diagonals of a rectangular parallelepiped are equal.
Pyramid is a polyhedron, the surface of which consists of a polygon - the base of the pyramid, and triangles that have a common vertex, called the lateral faces of the pyramid. The common vertex of these triangles is called top pyramids, ribs extending from the top, - lateral ribs pyramids.
The perpendicular dropped from the top of the pyramid to the base, as well as the length of this perpendicular, is called height pyramids.
The simplest pyramid - triangular or tetrahedron (Fig. 24.8). The peculiarity of a triangular pyramid is that any face can be considered as a base.
The pyramid is called correct, if its base is a regular polygon, and all side edges are equal to each other.
Note that we must distinguish regular tetrahedron(i.e. a tetrahedron in which all edges are equal to each other) and regular triangular pyramid(at its base lies a regular triangle, and the side edges are equal to each other, but their length may differ from the length of the side of the triangle, which is the base of the prism).
Distinguish bulging And non-convex polyhedra. You can define a convex polyhedron if you use the concept of a convex geometric body: a polyhedron is called convex. if it is a convex figure, i.e. together with any two of its points, it also entirely contains the segment connecting them.
A convex polyhedron can be defined differently: a polyhedron is called convex, if it lies entirely on one side of each of the polygons bounding it.
These definitions are equivalent. We do not provide proof of this fact.
All polyhedra that have been considered so far have been convex (cube, parallelepiped, prism, pyramid, etc.). The polyhedron shown in Fig. 24.9, is not convex.
It has been proven that in a convex polyhedron, all faces are convex polygons.
Let's consider several convex polyhedra (Table 24.1)
From this table it follows that for all considered convex polyhedra the equality B - P + G= 2. It turned out that this is also true for any convex polyhedron. This property was first proven by L. Euler and was called Euler's theorem.
A convex polyhedron is called correct if its faces are equal regular polygons and the same number of faces converge at each vertex.
Using the property of a convex polyhedral angle, one can prove that There are no more than five different types of regular polyhedra.
Indeed, if fan and polyhedron are regular triangles, then 3, 4 and 5 can converge at one vertex, since 60" 3< 360°, 60° - 4 < 360°, 60° 5 < 360°, но 60° 6 = 360°.
If three regular triangles converge at each vertex of a polyfan, then we get right-handed tetrahedron, which translated from Phetic means “tetrahedron” (Fig. 24.10, A).
If four regular triangles meet at each vertex of a polyhedron, then we get octahedron(Fig. 24.10, V). Its surface consists of eight regular triangles.
If five regular triangles converge at each vertex of a polyhedron, then we get icosahedron(Fig. 24.10, d). Its surface consists of twenty regular triangles.
If the faces of a polyfan are squares, then only three of them can converge at one vertex, since 90° 3< 360°, но 90° 4 = 360°. Этому условию удовлетворяет только куб. Куб имеет шесть фаней и поэтому называется также hexahedron(Fig. 24.10, b).
If the edges of a polyfan are regular pentagons, then only phi can converge at one vertex, since 108° 3< 360°, пятиугольники и в каждой вершине сходится три грани, называется dodecahedron(Fig. 24.10, d). Its surface consists of twelve regular pentagons.
The faces of a polyhedron cannot be hexagonal or more, since even for a hexagon 120° 3 = 360°.
In geometry, it has been proven that in three-dimensional Euclidean space there are exactly five different types of regular polyhedra.
To make a model of a polyhedron, you need to make it scan(more precisely, the development of its surface).
The development of a polyhedron is a figure on a plane that is obtained if the surface of the polyhedron is cut along certain edges and unfolded so that all the polygons included in this surface lie in the same plane.
Note that a polyhedron can have several different developments depending on which edges we cut. Figure 24.11 shows figures that are various developments of a regular quadrangular pyramid, i.e. a pyramid with a square at its base and all side edges equal to each other.
For a figure on a plane to be a development of a convex polyhedron, it must satisfy a number of requirements related to the features of the polyhedron. For example, the figures in Fig. 24.12 are not developments of a regular quadrangular pyramid: in the figure shown in Fig. 24.12, A, at the top M four faces converge, which cannot happen in a regular quadrangular pyramid; and in the figure shown in Fig. 24.12, b, lateral ribs A B And Sun not equal.
In general, the development of a polyhedron can be obtained by cutting its surface not only along the edges. An example of such a cube development is shown in Fig. 24.13. Therefore, more precisely, the development of a polyhedron can be defined as a flat polygon from which the surface of this polyhedron can be made without overlaps.
Bodies of revolution
Body of rotation called a body obtained as a result of the rotation of some figure (usually flat) around a straight line. This line is called axis of rotation.
Cylinder- ego body, which is obtained as a result of rotation of a rectangle around one of its sides. In this case, the specified party is axis of the cylinder. In Fig. 24.14 shows a cylinder with an axis OO', obtained by rotating a rectangle AA"O"O around a straight line OO". Points ABOUT And ABOUT"- centers of the cylinder bases.
A cylinder that results from rotating a rectangle around one of its sides is called straight circular a cylinder, since its bases are two equal circles located in parallel planes so that the segment connecting the centers of the circles is perpendicular to these planes. The lateral surface of the cylinder is formed by segments equal to the side of the rectangle parallel to the cylinder axis.
Sweep The lateral surface of a right circular cylinder, if cut along a generatrix, is a rectangle, one side of which is equal to the length of the generatrix, and the other to the length of the base circumference.
Cone- this is a body that is obtained as a result of rotation of a right triangle around one of the legs.
In this case, the indicated leg is motionless and is called the axis of the cone. In Fig. Figure 24.15 shows a cone with an axis SO, obtained by rotating a right triangle SOA with a right angle O around leg S0. Point S is called apex of the cone, OA- the radius of its base.
The cone that results from the rotation of a right triangle around one of its legs is called straight circular cone since its base is a circle, and its top is projected into the center of this circle. The lateral surface of the cone is formed by segments equal to the hypotenuse of the triangle, upon rotation of which a cone is formed.
If the side surface of the cone is cut along the generatrix, then it can be “unfolded” onto a plane. Sweep The lateral surface of a right circular cone is a circular sector with a radius equal to the length of the generatrix.
When a cylinder, cone or any other body of rotation intersects a plane containing the axis of rotation, it turns out axial section. The axial section of the cylinder is a rectangle, the axial section of the cone is an isosceles triangle.
Ball- this is a body that is obtained as a result of rotation of a semicircle around its diameter. In Fig. 24.16 shows a ball obtained by rotating a semicircle around the diameter AA". Full stop ABOUT called the center of the ball, and the radius of the circle is the radius of the ball.
The surface of the ball is called sphere. The sphere cannot be turned onto a plane.
Any section of a ball by a plane is a circle. The cross-sectional radius of the ball will be greatest if the plane passes through the center of the ball. Therefore, the section of a ball by a plane passing through the center of the ball is called large circle of the ball, and the circle that bounds it is large circle.
IMAGE OF GEOMETRIC BODIES ON THE PLANE
Unlike flat figures, geometric bodies cannot be accurately depicted, for example, on a sheet of paper. However, with the help of drawings on a plane, you can get a fairly clear image of spatial figures. To do this, special methods are used to depict such figures on a plane. One of them is parallel design.
Let a plane and a straight line intersecting a be given A. Let us take an arbitrary point A in space that does not belong to the line A, and we'll guide you through X direct A", parallel to the line A(Fig. 24.17). Straight A" intersects the plane at some point X", which is called parallel projection of point X onto plane a.
If point A lies on a straight line A, then with parallel projection X" is the point at which the line A intersects the plane A.
If the point X belongs to the plane a, then the point X" coincides with the point X.
Thus, if a plane a and a straight line intersecting it are given A. then each point X space can be associated with a single point A" - a parallel projection of the point X onto the plane a (when designing parallel to the straight line A). Plane A called projection plane. About the line A they say she will bark design direction - ggri replacement direct A any other direct design result parallel to it will not change. All lines parallel to a line A, specify the same design direction and are called along with the straight line A projecting straight lines.
Projection figures F call a set F' projection of all the points. Mapping each point X figures F"its parallel projection is a point X" figures F", called parallel design figures F(Fig. 24.18).
A parallel projection of a real object is its shadow falling on a flat surface in sunlight, since the sun's rays can be considered parallel.
Parallel design has a number of properties, knowledge of which is necessary when depicting geometric bodies on a plane. Let us formulate the main ones without providing their proof.
Theorem 24.1. During parallel design, the following properties are satisfied for straight lines not parallel to the design direction and for segments lying on them:
1) the projection of a line is a line, and the projection of a segment is a segment;
2) projections of parallel lines are parallel or coincide;
3) the ratio of the lengths of the projections of segments lying on the same line or on parallel lines is equal to the ratio of the lengths of the segments themselves.
From this theorem it follows consequence: with parallel projection, the middle of the segment is projected into the middle of its projection.
When depicting geometric bodies on a plane, it is necessary to ensure that the specified properties are met. Otherwise it can be arbitrary. Thus, the angles and ratios of the lengths of non-parallel segments can change arbitrarily, i.e., for example, a triangle in parallel design is depicted as an arbitrary triangle. But if the triangle is equilateral, then the projection of its median must connect the vertex of the triangle with the middle of the opposite side.
And one more requirement must be observed when depicting spatial bodies on a plane - to help create a correct idea of them.
Let us depict, for example, an inclined prism whose bases are squares.
Let's first build the lower base of the prism (you can start from the top). According to the rules of parallel design, oggo will be depicted as an arbitrary parallelogram ABCD (Fig. 24.19, a). Since the edges of the prism are parallel, we build parallel straight lines passing through the vertices of the constructed parallelogram and lay on them equal segments AA", BB', CC", DD", the length of which is arbitrary. By connecting points A", B", C", D in series ", we obtain a quadrilateral A" B "C" D", depicting the upper base of the prism. It is not difficult to prove that A"B"C"D"- parallelogram equal to parallelogram ABCD and, consequently, we have the image of a prism, the bases of which are equal squares, and the remaining faces are parallelograms.
If you need to depict a straight prism, the bases of which are squares, then you can show that the side edges of this prism are perpendicular to the base, as is done in Fig. 24.19, b.
In addition, the drawing in Fig. 24.19, b can be considered an image of a regular prism, since its base is a square - a regular quadrilateral, and also a rectangular parallelepiped, since all its faces are rectangles.
Let us now find out how to depict a pyramid on a plane.
To depict a regular pyramid, first draw a regular polygon lying at the base, and its center is a point ABOUT. Then draw a vertical segment OS depicting the height of the pyramid. Note that the verticality of the segment OS provides greater clarity of the drawing. Finally, point S is connected to all the vertices of the base.
Let us depict, for example, a regular pyramid, the base of which is a regular hexagon.
In order to correctly depict a regular hexagon during parallel design, you need to pay attention to the following. Let ABCDEF be a regular hexagon. Then ALLF is a rectangle (Fig. 24.20) and, therefore, during parallel design it will be depicted as an arbitrary parallelogram B"C"E"F". Since diagonal AD passes through point O - the center of the polygon ABCDEF and is parallel to the segments. BC and EF and AO = OD, then with parallel design it will be represented by an arbitrary segment A "D" , passing through the point ABOUT" parallel B"C" And E"F" and besides, A"O" = O"D".
Thus, the sequence of constructing the base of a hexagonal pyramid is as follows (Fig. 24.21):
§ depict an arbitrary parallelogram B"C"E"F" and its diagonals; mark the point of their intersection O";
§ through a point ABOUT" draw a straight line parallel V'S"(or E"F');
§ choose an arbitrary point on the constructed line A" and mark the point D" such that O"D" = A"O" and connect the dot A" with dots IN" And F", and point D" - with dots WITH" And E".
To complete the construction of the pyramid, draw a vertical segment OS(its length is chosen arbitrarily) and connect point S to all vertices of the base.
In parallel projection, the ball is depicted as a circle of the same radius. To make the image of the ball more visual, draw a projection of some large circle, the plane of which is not perpendicular to the projection plane. This projection will be an ellipse. The center of the ball will be represented by the center of this ellipse (Fig. 24.22). Now we can find the corresponding poles N and S, provided that the segment connecting them is perpendicular to the equatorial plane. To do this, through the point ABOUT draw a straight line perpendicular AB and mark point C - the intersection of this line with the ellipse; then through point C we draw a tangent to the ellipse representing the equator. It has been proven that the distance CM equal to the distance from the center of the ball to each of the poles. Therefore, putting aside the segments ON And OS equal CM, we get the poles N and S.
Let's consider one of the techniques for constructing an ellipse (it is based on a transformation of the plane, which is called compression): construct a circle with a diameter and draw chords perpendicular to the diameter (Fig. 24.23). Half of each chord is divided in half and the resulting points are connected by a smooth curve. This curve is an ellipse whose major axis is the segment AB, and the center is a point ABOUT.
This technique can be used to depict a straight circular cylinder (Fig. 24.24) and a straight circular cone (Fig. 24.25) on a plane.
A straight circular cone is depicted like this. First, they build an ellipse - the base, then find the center of the base - the point ABOUT and draw a line segment perpendicularly OS which represents the height of the cone. From point S, tangents are drawn to the ellipse (this is done “by eye”, applying a ruler) and segments are selected SC And SD these straight lines from point S to points of tangency C and D. Note that the segment CD does not coincide with the diameter of the base of the cone.
1 option
1. A body whose surface consists of a finite number of flat polygons is called:
1. Quadrilateral 2. Polygon 3. Polyhedron 4. Hexagon
2. Polyhedra include:
1. Parallelepiped 2. Prism 3. Pyramid 4. All answers are correct
3. A segment connecting two vertices of a prism that do not belong to the same face is called:
1. Diagonal 2. Edge 3. Face 4. Axis
4. The prism has side ribs:
1. Equal 2. Symmetrical 3. Parallel and equal 4. Parallel
5. The faces of a parallelepiped that do not have common vertices are called:
1. Opposite 2. Opposite 3. Symmetrical 4. Equal
6. A perpendicular dropped from the top of the pyramid to the plane of the base is called:
1. Median 2. Axis 3. Diagonal 4. Height
7. Points that do not lie in the plane of the base of the pyramid are called:
1. Tops of the pyramid 2. Lateral ribs 3. Linear size
4. Vertices of the face
8. The height of the side face of a regular pyramid drawn from its vertex is called:
1. Median 2. Apothem 3. Perpendicular 4. Bisector
9. The cube has all the faces:
1. Rectangles 2. Squares 3. Trapezes 4. Rhombuses
10. A body consisting of two circles and all segments connecting the points of the circles is called:
1. Cone 2. Ball 3. Cylinder 4. Sphere
11. The cylinder has generators:
1. Equal 2. Parallel 3. Symmetrical 4. Parallel and equal
12. The bases of the cylinder lie in:
1. Same plane 2. Equal planes 3. Parallel planes 4. Different planes
13. The surface of the cone consists of:
1. Generators 2. Faces and edges 3. Bases and edges 4. Bases and side surfaces
14. A segment connecting two points of a spherical surface and passing through the center of the ball is called:
1. Radius 2. Center 3. Axis 4. Diameter
15. Every section of a ball by a plane is:
1. Circle 2. Circle 3. Sphere 4. Semicircle
16. The section of a ball by the diametrical plane is called:
1. Large circle 2. Large circle 3. Small circle 4. Circle
17. The circle of a cone is called:
1. Top 2. Plane 3. Face 4. Base
18. Prism bases:
1. Parallel 2. Equal 3. Perpendicular 4. Not equal
19. The lateral surface area of the prism is called:
1. Sum of areas of lateral polygons
2. Sum of areas of lateral ribs
3. Sum of areas of lateral faces
4. Sum of base areas
20. The intersection of the diagonals of a parallelepiped is its:
1. Center 2. Center of symmetry 3. Linear dimension 4. Section point
21. The radius of the base of the cylinder is 1.5 cm, height is 4 cm. Find the diagonal of the axial section.
1. 4.2 cm. 2. 10 cm. 3. 5 cm.
0 . What is the diameter of the base if the generatrix is 7 cm?
1. 7 cm. 2. 14 cm. 3. 3.5 cm.
23. The height of the cylinder is 8 cm, the radius is 1 cm. Find the area of the axial section.
1.9 cm 2 . 2.8 cm 2 3. 16 cm 2 .
24. The radii of the bases of a truncated cone are 15 cm and 12 cm, height 4 cm. What is the generatrix of the cone?
1. 5 cm 2. 4 cm 3. 10 cm
POLYHEDRONS AND BODIES OF ROTATION
Option 2
1. The vertices of the polyhedron are designated:
1. a, b, c, d... 2. A, B, C, D ... 3. ab, CD, ac, ad... 4. AB, SV, A D, CD...
2. A polyhedron that consists of two flat polygons combined by parallel translation is called:
1. Pyramid 2. Prism 3. Cylinder 4. Parallelepiped
3. If the lateral edges of the prism are perpendicular to the base, then the prism is:
1. Oblique 2. Regular 3. Straight 4. Convex
4. If a parallelogram lies at the base of a prism, then it is:
1. Regular prism 2. Parallelepiped 3. Regular polygon
4. Pyramid
5. A polyhedron, which consists of a flat polygon, a point and segments connecting them, is called:
1. Cone 2. Pyramid 3. Prism 4. Ball
6. The segments connecting the top of the pyramid with the vertices of the base are called:
1. Edges 2. Sides 3. Side edges 4. Diagonals
7. A triangular pyramid is called:
1. Regular pyramid 2. Tetrahedron 3. Triangular pyramid 4. Inclined pyramid
8. The following does not apply to regular polyhedra:
1. Cube 2. Tetrahedron 3. Icosahedron 4. Pyramid
9. The height of the pyramid is:
1. Axis 2. Median 3. Perpendicular 4. Apothema
10. The segments connecting the points of the circles’ circumferences are called:
1. Faces of the cylinder 2. Generics of the cylinder 3. Heights of the cylinder
4. Perpendiculars of the cylinder
1. Cylinder axis 2. Cylinder height 3. Cylinder radius
4. Cylinder rib
12. A body that consists of a point, a circle and segments connecting them is called:
1. Pyramid 2. Cone 3. Sphere 4. Cylinder
13. A body that consists of all points in space is called:
1. Sphere 2. Ball 3. Cylinder 4. Hemisphere
14. The boundary of the ball is called:
1. Sphere 2. Ball 3. Section 4. Circle
15. The line of intersection of two spheres is:
1. Circle 2. Semicircle 3. Circle 4. Section
16. The section of a sphere is called:
1. Circle 2. Large circle 3. Small circle 4. Small circle
17. The faces of a convex polyhedron are convex:
1. Triangles 2. Angles 3. Polygons 4. Hexagons
18. The lateral surface of the prism consists of...
1. Parallelograms 2. Squares 3. Diamonds 4. Triangles
19. The lateral surface of a straight prism is equal to:
1. Product of the perimeter and the length of the prism face
2. The product of the length of the prism face and the base
3. Product of the length of the prism face and the height
4. Product of the perimeter of the base and the height of the prism
20. Regular polyhedra include:
21. The radius of the base of the cylinder is 2.5 cm, height is 12 cm. Find the diagonal of the axial section.
1. 15 cm; 2. 14 cm; 3. 13 cm.
22. The largest angle between the generatrices of the cone is 60 0 . What is the diameter of the base if the generatrix is 5 cm?
1.5 cm; 2. 10 cm; 3. 2.5 cm.
23. The height of the cylinder is 4 cm, the radius is 1 cm. Find the area of the axial section.
1.9 cm 2 . 2.8 cm 2 3. 16 cm 2 .
24. The radii of the bases of a truncated cone are 6 cm and 12 cm, height 8 cm. What is the generatrix of the cone?
1. 10 cm; 2.4 cm; 3.6 cm.
Geometric bodies
Introduction
In stereometry, figures in space are studied, which are called geometric bodies.
The objects around us give us an idea of geometric bodies. Unlike real objects, geometric bodies are imaginary objects. Clearly geometric body one must imagine it as a part of space occupied by matter (clay, wood, metal, ...) and limited by a surface.
All geometric bodies are divided into polyhedra And round bodies.
Polyhedra
Polyhedron is a geometric body whose surface consists of a finite number of flat polygons.
Edges polyhedron, the polygons that make up its surface are called.
Ribs of a polyhedron, the sides of the faces of the polyhedron are called.
Peaks of a polyhedron are called the vertices of the faces of the polyhedron.
Polyhedra are divided into convex And non-convex.
The polyhedron is called convex, if it lies entirely on one side of any of its faces.
Exercise. Specify edges, ribs And peaks cube shown in the figure.
Convex polyhedra are divided into prisms And pyramids.
Prism
Prism is a polyhedron in which two faces are equal and parallel
n-gons, and the rest n faces are parallelograms.
Two n-gons are called prism bases, parallelograms – side faces. The sides of the side faces and bases are called prism ribs, the ends of the edges are called the vertices of the prism. Side edges are edges that do not belong to the bases.
Polygons A 1 A 2 ...A n and B 1 B 2 ...B n are the bases of the prism.
Parallelograms A 1 A 2 B 2 B 1, ... - side faces.
Prism properties:
· The bases of the prism are equal and parallel.
· The lateral edges of the prism are equal and parallel.
Prism diagonal called a segment connecting two vertices that do not belong to the same face.
Prism height is called a perpendicular dropped from a point of the upper base to the plane of the lower base.
A prism is called 3-gonal, 4-gonal, ..., n-coal, if its base
3-gons, 4-gons, ..., n-gons.
Direct prism called a prism whose side ribs are perpendicular to the bases. The lateral faces of a straight prism are rectangles.
Inclined prism called a prism that is not straight. The lateral faces of an inclined prism are parallelograms.
With the right prism called straight a prism with regular polygons at its base.
Area full surface prisms is called the sum of the areas of all its faces.
Area lateral surface prisms is called the sum of the areas of its lateral faces.
S full = S side + 2 S basic
Polyhedron
Polyhedron- this is a body whose surface consists of a finite number of flat polygons.
The polyhedron is called convex
The polyhedron is called convex ,if it is located on one side of each flat polygon on its surface.
Euclid (presumably 330-277 BC) - mathematician of the Alexandrian school of Ancient Greece, author of the first treatise on mathematics that has come down to us, “Elements” (in 15 books)
side faces.
A prism is a polyhedron, which consists of two flat polygons lying in different planes and combined by parallel translation, and all the segments connecting the corresponding points of these polygons. Polygons Ф and Ф1 lying in parallel planes are called prism bases, and the remaining faces are called side faces.
The surface of the prism thus consists of two equal polygons (bases) and parallelograms (side faces). There are triangular, quadrangular, pentagonal, etc. prisms. depending on the number of vertices of the base.
If the lateral edge of a prism is perpendicular to the plane of its base, then such a prism is called straight ; if the lateral edge of the prism is not perpendicular to the plane of its base, then such a prism is called inclined . A straight prism has rectangular side faces.
The bases of the prism are equal.
The bases of the prism are equal.
The bases of a prism lie in parallel planes.
The side edges of a prism are parallel and equal.
The height of a prism is the distance between the planes of its bases.
It turns out that a prism can be not only a geometric body, but also an artistic masterpiece. It was the prism that became the basis for the paintings of Picasso, Braque, Griss, etc.
It turns out that a snowflake can take the shape of a hexagonal prism, but this will depend on the air temperature.
In the 3rd century BC. e. a lighthouse was built so that ships could safely pass the reefs on their way to Alexandria Bay. At night they were helped in this by the reflection of flames, and during the day by a column of smoke. It was the world's first lighthouse, and it stood for 1,500 years.
The lighthouse was built on the small island of Pharos in the Mediterranean Sea, off the coast of Alexandria. It took 20 years to build and was completed around 280 BC.
In the 14th century, the lighthouse was destroyed by an earthquake. Its debris was used in the construction of a military fort. The fort has been rebuilt several times and still stands on the site of the world's first lighthouse.
Mausolus was the ruler of Caria. The capital of the region was Halicarnassus. Mausolus married his sister Artemisia. He decided to build a tomb for himself and his queen. Mavsol dreamed of a majestic monument that would remind the world of his wealth and power. He died before work on the tomb was completed. Artemisia continued to lead the construction. The tomb was built in 350 BC. e. It was named Mausoleum after the king.
The ashes of the royal couple were kept in golden urns in a tomb at the base of the building. A row of stone lions guarded this room. The structure itself resembled a Greek temple, surrounded by columns and statues. At the top of the building was a step pyramid. At a height of 43 m above the ground, it was crowned with a sculpture of a chariot drawn by horses. There were probably statues of the king and queen on it.
Eighteen centuries later, an earthquake destroyed the Mausoleum to the ground. Another three hundred years passed before archaeologists began excavations. In 1857, all the finds were transported to the British Museum in London. Now, in the place where the Mausoleum once was, only a handful of stones remain.
crystals.
There are not only geometric shapes created by human hands. There are many of them in nature itself. The impact on the appearance of the earth’s surface of such natural factors as wind, water, sunlight is very spontaneous and chaotic. However, sand dunes, pebbles on the seashore, The crater of an extinct volcano, as a rule, has geometrically regular shapes. Sometimes stones are found in the ground of such a shape, as if someone had carefully cut them out, ground them, and polished them. This is - crystals.
parallelepiped.
If the base of the prism is a parallelogram, then it is called parallelepiped.
The models of a rectangular parallelepiped are:
cool room
It turns out that calcite crystals, no matter how much they are crushed into smaller parts, always break up into fragments shaped like a parallelepiped.
City buildings most often have the shape of polyhedra. As a rule, these are ordinary parallelepipeds. And only unexpected architectural solutions decorate cities.
1. Is a prism regular if its edges are equal?
a) yes; c) no. Justify your answer.
2. The height of a regular triangular prism is 6 cm. The side of the base is 4 cm. Find the total surface area of this prism.
3. The areas of the two lateral faces of an inclined triangular prism are 40 and 30 cm2. The angle between these faces is straight. Find the lateral surface area of the prism.
4. In the parallelepiped ABCDA1B1C1D1, sections A1BC and CB1D1 are drawn. In what ratio do these planes divide diagonal AC1?
1) a tetrahedron with 4 faces, 4 vertices, 6 edges;
2) cube - 6 faces, 8 vertices, 12 edges;
3) octahedron - 8 faces, 6 vertices, 12 edges;
4) dodecahedron - 12 faces, 20 vertices, 30 edges;
5) icosahedron - 20 faces, 12 vertices, 30 edges.
Thales of Miletus, founder Ionian Pythagoras of Samos
Scientists and philosophers of Ancient Greece adopted and reworked the achievements of culture and science of the Ancient East. Thales, Pythagoras, Democritus, Eudoxus and others traveled to Egypt and Babylon to study music, mathematics and astronomy. It is no coincidence that the beginnings of Greek geometric science are associated with the name Thales of Miletus, founder Ionian schools. The Ionians, who inhabited the territory that bordered the eastern countries, were the first to borrow the knowledge of the East and began to develop it. Scientists of the Ionian school were the first to subject to logical processing and systematize mathematical information borrowed from the ancient Eastern peoples, especially from the Babylonians. Proclus and other historians attribute many geometric discoveries to Thales, the head of this school. About attitude Pythagoras of Samos to geometry, Proclus writes the following in his commentary to Euclid’s Elements: “He studied this science (i.e., geometry), starting from its first foundations, and tried to obtain theorems using purely logical thinking.” Proclus attributes to Pythagoras, in addition to the well-known theorem on the square of the hypotenuse, the construction of five regular polyhedra:
Plato's solids
Plato's solids are convex polyhedra, all of whose faces are regular polygons. All polyhedral angles of a regular polyhedron are congruent. As follows from calculating the sum of plane angles at a vertex, there are no more than five convex regular polyhedra. Using the method indicated below, one can prove that there are exactly five regular polyhedra (this was proven by Euclid). They are regular tetrahedron, cube, octahedron, dodecahedron and icosahedron.
Octahedron (Fig. 3).
Octahedron -octahedron; a body bounded by eight triangles; a regular octahedron is bounded by eight equilateral triangles; one of the five regular polyhedra. (Fig. 3).
Dodecahedron -dodecahedron, a body bounded by twelve polygons; regular pentagon; one of the five regular polyhedra . (Fig. 4).
Icosahedron -twenty-hedron, a body bounded by twenty polygons; the regular icosahedron is limited by twenty equilateral triangles; one of the five regular polyhedra. (Fig. 5).
Reliable information about the life and scientific activities of Pythagoras has not been preserved. He is credited with creating the doctrine of the similarity of figures. He was probably among the first scientists to view geometry not as a practical and applied discipline, but as an abstract logical science.
The faces of the dodecahedron are regular pentagons. The diagonals of a regular pentagon form the so-called star pentagon - a figure that served as an emblem, an identification mark for the students of Pythagoras. It is known that the Pythagorean League was at the same time a philosophical school, a political party and a religious brotherhood. According to legend, one Pythagorean fell ill in a foreign land and could not pay the owner of the house who cared for him before his death. The latter painted a star-shaped pentagon on the wall of his house. Seeing this sign a few years later, another wandering Pythagorean inquired about what had happened from the owner and generously rewarded him.
The school of Pythagoras discovered the existence of incommensurable quantities, that is, those whose relationship cannot be expressed by any integer or fractional number. An example is the ratio of the length of the diagonal of a square to the length of its side, equal to C2. This number is not rational (i.e., an integer or a ratio of two integers) and is called irrational, i.e. irrational (from the Latin ratio - attitude).
Tetrahedron (Fig. 1).
Tetrahedron -tetrahedron, all faces of which are triangles, i.e. triangular pyramid; a regular tetrahedron is bounded by four equilateral triangles; one of the five regular polygons. (Fig. 1).
Cube or regular hexahedron (Fig. 2).
Tetrahedron -tetrahedron, all faces of which are triangles, i.e. triangular pyramid; a regular tetrahedron is bounded by four equilateral triangles; one of the five regular polygons. (Fig. 1).
Tetrahedron -tetrahedron, all faces of which are triangles, i.e. triangular pyramid; a regular tetrahedron is bounded by four equilateral triangles; one of the five regular polygons. (Fig. 1).
Cube or regular hexahedron - a regular quadrangular prism with equal edges, limited by six squares. (Fig. 2).
Pyramid
Pyramid- a polyhedron, which consists of a flat polygon - the base of the pyramid, points that do not lie in the plane of the base-top of the pyramid and all segments connecting the top of the pyramid with the points of the base
Definition . A pyramid whose base is a regular polygon and whose vertex is projected into its center is called regular.
The figure shows a regular hexagonal pyramid.
The picture shows a pentagonal pyramid SABCDE and its development. Triangles that have a common vertex are called side faces pyramids; common vertex of the side faces - top pyramids; a polygon to which this vertex does not belong is basis pyramids; the edges of the pyramid converging at its apex - lateral ribs pyramids. Height pyramid is a perpendicular segment drawn through its top to the base plane, with ends at the top and on the base plane of the pyramid. In the figure there is a segment SO- height of the pyramid.
Among the remarkable Greek scientists of the V - IV centuries. BC, who developed the theory of volumes were Democritus of Abdera and Eudoxus of Cnidus.
Euclid does not use the term "volume". For him, the term “cube,” for example, also means the volume of a cube. In Book XI of the "Principles" the following theorems are presented, among others.
1. Parallelepipeds with equal heights and equal bases are equal in size.
2. The ratio of the volumes of two parallelepipeds with equal heights is equal to the ratio of the areas of their bases.
3. In parallelepipeds of equal area, the areas of the bases are inversely proportional to the heights.
Euclid's theorems relate only to the comparison of volumes, since Euclid probably considered the direct calculation of the volumes of bodies to be a matter of practical manuals in geometry. In the applied works of Heron of Alexandria, there are rules for calculating the volume of a cube, prism, parallelepiped and other spatial figures.
The volumes of grain barns and other structures in the form of cubes, prisms and cylinders were calculated by the Egyptians and Babylonians, the Chinese and Indians by multiplying the base area by the height. However, the ancient East knew mainly only certain rules, found experimentally, which were used to find volumes for the areas of figures. At a later time, when geometry was formed as a science, a general approach to calculating the volumes of polyhedra was found.
A prism whose base is a parallelogram is called a parallelepiped.
According to the definition a parallelepiped is a quadrangular prism, all of whose faces are parallelograms. Parallelepipeds, like prisms, can be straight And inclined. Figure 1 shows an inclined parallelepiped, and Figure 2 shows a straight parallelepiped.
A right parallelepiped whose base is a rectangle is called rectangular parallelepiped. All faces of a rectangular parallelepiped are rectangles. Models of a rectangular parallelepiped are a classroom, a brick, and a matchbox.
The lengths of three edges of a rectangular parallelepiped having a common end are called measurements. For example, there are matchboxes with dimensions of 15, 35, 50 mm. A cube is a rectangular parallelepiped with equal dimensions. All six faces of the cube are equal squares.
Let's consider some properties of a parallelepiped.
Theorem. The parallelepiped is symmetrical about the middle of its diagonal.
It follows directly from the theorem important properties of a parallelepiped:
1. Any segment with ends belonging to the surface of the parallelepiped and passing through the middle of its diagonal is divided in half by it; in particular, all diagonals of a parallelepiped intersect at one point and are bisected by it. 2. Opposite faces of a parallelepiped are parallel and equal
