How many vertices does a hexagonal pyramid have. Geometric figures

Here are collected basic information about the pyramids and related formulas and concepts. All of them are studied with a tutor in mathematics in preparation for the exam.

Consider a plane, a polygon lying in it and a point S not lying in it. Connect S to all vertices of the polygon. The resulting polyhedron is called a pyramid. The segments are called lateral edges. The polygon is called the base, and the point S is called the top of the pyramid. Depending on the number n, the pyramid is called triangular (n=3), quadrangular (n=4), pentagonal (n=5) and so on. Alternative name for the triangular pyramid - tetrahedron. The height of a pyramid is the perpendicular drawn from its apex to the base plane.

A pyramid is called correct if a regular polygon, and the base of the height of the pyramid (the base of the perpendicular) is its center.

Tutor's comment:
Do not confuse the concept of "regular pyramid" and "regular tetrahedron". In a regular pyramid, the side edges are not necessarily equal to the edges of the base, but in a regular tetrahedron, all 6 edges of the edges are equal. This is his definition. It is easy to prove that the equality implies that the center P of the polygon with a height base, so a regular tetrahedron is a regular pyramid.

What is an apothem?
The apothem of a pyramid is the height of its side face. If the pyramid is regular, then all its apothems are equal. The reverse is not true.

Mathematics tutor about his terminology: work with pyramids is 80% built through two types of triangles:
1) Containing apothem SK and height SP
2) Containing the lateral edge SA and its projection PA

To simplify references to these triangles, it is more convenient for a math tutor to name the first of them apothemic, and second costal. Unfortunately, you will not find this terminology in any of the textbooks, and the teacher has to introduce it unilaterally.

Pyramid volume formula:
1) , where is the area of ​​the base of the pyramid, and is the height of the pyramid
2) , where is the radius of the inscribed sphere, and is the total surface area of ​​the pyramid.
3) , where MN is the distance of any two crossing edges, and is the area of ​​the parallelogram formed by the midpoints of the four remaining edges.

Pyramid Height Base Property:

Point P (see figure) coincides with the center of the inscribed circle at the base of the pyramid if one of the following conditions is met:
1) All apothems are equal
2) All side faces are equally inclined towards the base
3) All apothems are equally inclined to the height of the pyramid
4) The height of the pyramid is equally inclined to all side faces

Math tutor's commentary: note that all points are united by one common property: one way or another, side faces participate everywhere (apothems are their elements). Therefore, the tutor can offer a less precise, but more convenient formulation for memorization: the point P coincides with the center of the inscribed circle, the base of the pyramid, if there is any equal information about its lateral faces. To prove it, it suffices to show that all apothemical triangles are equal.

The point P coincides with the center of the circumscribed circle near the base of the pyramid, if one of the three conditions is true:
1) All side edges are equal
2) All side ribs are equally inclined towards the base
3) All side ribs are equally inclined to the height

This lesson provides the definition and properties of a regular triangular pyramid and its special case - a tetrahedron (see below). Links to examples of problem solving are provided at the end of the lesson.

Definition

Regular triangular pyramid- This is a pyramid, the base of which is a regular triangle, and the top is projected into the center of the base.

The figure shows:
ABC- Base pyramids
OS - Height
KS - Apothem
OK - radius of the circle inscribed in the base
AO - radius of a circle circumscribed around the base of a regular triangular pyramid
SKO - the dihedral angle between the base and the face of the pyramid (they are equal in a regular pyramid)

Important. In a regular triangular pyramid, the length of the edge (in the figure AS, BS, CS) may not be equal to the length of the side of the base (in the figure AB, AC, BC). If the length of the edge of a regular triangular pyramid is equal to the length of the side of the base, then such a pyramid is called a tetrahedron (see below).

Properties of a regular triangular pyramid:

• lateral edges of a regular pyramid are equal
• all side faces of a regular pyramid are isosceles triangles
• in a regular triangular pyramid, you can both inscribe and describe a sphere around it
• if the centers of the spheres inscribed and circumscribed around a regular triangular pyramid coincide, then the sum of the plane angles at the top of the pyramid is equal to π (180 degrees), and each of them, respectively, is equal to π / 3 (pi divided by 3 or 60 degrees).
• the area of ​​the lateral surface of a regular pyramid is equal to half the product of the perimeter of the base and the apothem
• the top of the pyramid is projected onto the base at the center of a regular equilateral triangle, which is the center of the inscribed circle and the intersection point of the medians

Formulas for a regular triangular pyramid

The formula for the volume of a regular triangular pyramid is:

V is the volume of a regular pyramid with a regular (equilateral) triangle at the base
h - the height of the pyramid
a - the length of the side of the base of the pyramid
R - radius of the circumscribed circle
r - radius of the inscribed circle

Since a regular triangular pyramid is a special case of a regular pyramid, the formulas that are true for a regular pyramid are also true for a regular triangular pyramid - see formulas for a regular pyramid.

Examples of problem solving:

Tetrahedron

A special case of a regular triangular pyramid is tetrahedron.

Tetrahedron- this is a regular polyhedron (regular triangular pyramid) in which all faces are regular triangles.

At the tetrahedron:

• All edges are equal
• 4 faces, 4 vertices and 6 edges
• All dihedral angles at edges and all trihedral angles at vertices are equal

Median of a tetrahedron- this is a segment connecting the vertex to the point of intersection of the medians of the opposite face (the medians of an equilateral triangle opposite the vertex)

Bimedian tetrahedron- this is a segment connecting the midpoints of crossing edges (connecting the midpoints of the sides of a triangle, which is one of the faces of a tetrahedron)

Tetrahedron height- this is a segment connecting the vertex with a point of the opposite face and perpendicular to this face (that is, it is the height drawn from any face, also coincides with the center of the circumscribed circle).

Tetrahedron has the following properties:

• All medians and bimedians of a tetrahedron intersect at one point
• This point divides the medians in a ratio of 3:1, counting from the top
• This point bisects the bimedians

Chapter 1. Theoretical study of the types of sections and methods for their construction in a regular quadrangular pyramid

Pyramid (ancient Greek Πυραμίς, genus P. πυραμίδος) is a polyhedron whose base is a polygon, and the remaining faces are triangles with a common vertex. By the number of corners of the base, pyramids are triangular, quadrangular, etc. The pyramid is a special case of a cone.

The beginning of the geometry of the pyramid was laid in ancient Egypt and Babylon, but it was actively developed in ancient Greece. The first to establish what the volume of the pyramid is equal to was Democritus, and Eudoxus of Cnidus proved it. The ancient Greek mathematician Euclid systematized knowledge about the pyramid in the XII volume of his "Beginnings", and also brought out the first definition of the pyramid: a solid figure bounded by planes that converge from one plane at one point.

pyramid elements

apothem - the height of the side face of a regular pyramid, drawn from its top;

side faces - triangles converging at the top of the pyramid;

side edges - common sides of the side faces;

The top of the pyramid is a point connecting the side edges and not lying in the plane of the base;

height - a segment of a perpendicular drawn through the top of the pyramid to the plane of its base (the ends of this segment are the top of the pyramid and the base of the perpendicular);

Diagonal section of a pyramid - a section of a pyramid passing through the top and the diagonal of the base;

base - a polygon that does not belong to the top of the pyramid.

Pyramid properties:

The number of faces of the pyramid is equal to its number of vertices.

Any polyhedron with the same number of faces as the number of vertices is a pyramid. The total number of vertices in the pyramid is n+1, where n is the number of vertices at the base.

If all side edges are equal, then:

§ near the base of the pyramid, a circle can be described, and the top of the pyramid is projected into its center;

§ side ribs form equal angles with the base plane.

§ The converse is also true, that is, if the side edges form equal angles with the base plane, or if a circle can be described near the base of the pyramid, and the top of the pyramid is projected into its center, then all the side edges of the pyramid are equal.

If the side faces are inclined to the base plane at one angle, then:

§ a circle can be inscribed in the base of the pyramid, and the top of the pyramid is projected into its center;

§ the heights of the side faces are equal;

§ The area of ​​the lateral surface is equal to half the product of the perimeter of the base and the height of the lateral face.

Types of sections in a regular quadrangular pyramid:

Diagonal section of a pyramid

• apothem- the height of the side face of a regular pyramid, which is drawn from its top (in addition, the apothem is the length of the perpendicular, which is lowered from the middle of a regular polygon to 1 of its sides);
• side faces (ASB, BSC, CSD, DSA) - triangles that converge at the top;
• side ribs ( AS , BS , CS , D.S. ) - common sides of the side faces;
• top of the pyramid (v. S) - a point that connects the side edges and which does not lie in the plane of the base;
• height ( SO ) - a segment of the perpendicular, which is drawn through the top of the pyramid to the plane of its base (the ends of such a segment will be the top of the pyramid and the base of the perpendicular);
• diagonal section of a pyramid- section of the pyramid, which passes through the top and the diagonal of the base;
• base (ABCD) is a polygon to which the top of the pyramid does not belong.

pyramid properties.

1. When all side edges are the same size, then:

• near the base of the pyramid it is easy to describe a circle, while the top of the pyramid will be projected into the center of this circle;
• side ribs form equal angles with the base plane;
• in addition, the converse is also true, i.e. when the side edges form equal angles with the base plane, or when a circle can be described near the base of the pyramid and the top of the pyramid will be projected into the center of this circle, then all the side edges of the pyramid have the same size.

2. When the side faces have an angle of inclination to the plane of the base of the same value, then:

• near the base of the pyramid, it is easy to describe a circle, while the top of the pyramid will be projected into the center of this circle;
• the heights of the side faces are of equal length;
• the area of ​​the side surface is ½ the product of the perimeter of the base and the height of the side face.

3. A sphere can be described near the pyramid if the base of the pyramid is a polygon around which a circle can be described (a necessary and sufficient condition). The center of the sphere will be the point of intersection of the planes that pass through the midpoints of the edges of the pyramid perpendicular to them. From this theorem we conclude that a sphere can be described both around any triangular and around any regular pyramid.

4. A sphere can be inscribed in a pyramid if the bisector planes of the internal dihedral angles of the pyramid intersect at the 1st point (a necessary and sufficient condition). This point will become the center of the sphere.

The simplest pyramid.

According to the number of corners of the base of the pyramid, they are divided into triangular, quadrangular, and so on.

The pyramid will triangular, quadrangular, and so on, when the base of the pyramid is a triangle, a quadrilateral, and so on. A triangular pyramid is a tetrahedron - a tetrahedron. Quadrangular - pentahedron and so on.