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A triangular pyramid whose edges are equal. The main properties of the correct pyramid

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 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". At the right pyramid side ribs 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

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 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.

Video lesson 2: Pyramid challenge. Pyramid Volume

Video lesson 3: Pyramid challenge. Correct pyramid

Lecture: Pyramid, its base, lateral edges, height, lateral surface; triangular pyramid; right pyramid

Pyramid, its properties

Pyramid- This is a three-dimensional body that has a polygon at the base, and all its faces consist of triangles.

A special case of a pyramid is a cone, at the base of which lies a circle.

Consider the main elements of the pyramid:

Apothem is a segment that connects the top of the pyramid with the middle of the lower edge of the side face. In other words, this is the height of the face of the pyramid.

In the figure you can see the triangles ADS, ABS, BCS, CDS. If you look closely at the names, you can see that each triangle has one common letter in its name - S. That is, this means that all side faces (triangles) converge at one point, which is called the top of the pyramid.

The segment OS, which connects the vertex with the point of intersection of the diagonals of the base (in the case of triangles, at the point of intersection of the heights), is called pyramid height.

A diagonal section is a plane that passes through the top of the pyramid, as well as one of the diagonals of the base.

Since the lateral surface of the pyramid consists of triangles, to find the total area of the lateral surface, it is necessary to find the areas of each face and add them. The number and shape of the faces depends on the shape and size of the sides of the polygon that lies at the base.

The only plane in a pyramid that does not have a vertex is called basis pyramids.

In the figure, we see that the base is a parallelogram, however, there can be any arbitrary polygon.

Properties:

Consider the first case of a pyramid, in which it has edges of the same length:

A circle can be described around the base of such a pyramid. If you project the top of such a pyramid, then its projection will be located in the center of the circle.
The angles at the base of the pyramid are the same for each face.
At the same time, a sufficient condition for the fact that a circle can be described around the base of the pyramid, and also that all the edges are of different lengths, can be considered the same angles between the base and each edge of the faces.

If you come across a pyramid in which the angles between the side faces and the base are equal, then the following properties are true:

You will be able to describe a circle around the base of the pyramid, the top of which is projected exactly to the center.
If you draw at each side face of the height to the base, then they will be of equal length.
To find the lateral surface area of such a pyramid, it is enough to find the perimeter of the base and multiply it by half the length of the height.
Sbp \u003d 0.5P oc H.
Types of pyramid.
Depending on which polygon lies at the base of the pyramid, they can be triangular, quadrangular, etc. If a regular polygon (with equal sides) lies at the base of the pyramid, then such a pyramid will be called regular.

Regular triangular pyramid

We are well aware of the great Egyptian pyramids, everyone can imagine what they look like. This representation will help us understand the features of such geometric figure like a pyramid.

A pyramid is a polyhedron consisting of a flat polygon - the base of the pyramid, a point that does not lie in the plane of the base - the top of the pyramid and all segments connecting the top with the points of the base. The segments that connect the top of the pyramid with the top of the base are called lateral edges. On fig. 1 shows the pyramid SABCD. Quadrilateral ABCD is the base of the pyramid, point S is the top of the pyramid, segments SA, SB, SC and SD are the edges of the pyramid.

The height of the pyramid is the perpendicular dropped from the top of the pyramid to the plane of the base. On fig. 1 SO is the height of the pyramid.

A pyramid is called n-gonal if its base is an n-gon. Figure 1 shows a quadrangular pyramid. A triangular pyramid is called a tetrahedron.

A pyramid is called regular if its base is a regular polygon, and the base of the height coincides with the center of this polygon. The lateral edges of a regular pyramid are equal, and, therefore, the lateral faces are isosceles triangles. In a regular pyramid, the height of the side face drawn from the top of the pyramid is called apothem.

The pyramid has a number of properties.

All diagonals of a pyramid belong to its faces.

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;
the side edges form equal angles with the base plane, and, conversely, 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 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 at 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.

Consider the formulas for finding the volume, surface area of the pyramid.

The volume of the pyramid can be calculated using the following formula:

where S is the area of the base and h is the height.

To find the total surface area of a pyramid, use the formula:

S p \u003d S b + S o,

where S p is the total surface area, S b is the side surface area, S o is the base area.

A truncated pyramid is a polyhedron enclosed between the base of the pyramid and a cutting plane parallel to its base. The faces of the truncated pyramid, lying in parallel planes, are called the bases of the truncated pyramid, the remaining faces are called the side faces. The bases of a truncated pyramid are similar polygons, the side faces are trapezoids. A truncated pyramid that is obtained from a regular pyramid is called a regular truncated pyramid. The side faces of a regular truncated trapezoid are equal isosceles trapezoids, their heights are called apothems.