home Botany book Geometric figures. Pyramid. Correct pyramid. Definition Correct 4 sided pyramid formula

Geometric figures. Pyramid. Correct pyramid. Definition Correct 4 sided pyramid formula

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 ribs form with the base plane equal angles, 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, which means that all 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.

Definition 1. A pyramid is called regular if its base is a regular polygon, and the top of such a pyramid is projected into the center of its base.

Definition 2. A pyramid is called regular if its base is a regular polygon and its height passes through the center of the base.

Elements of a regular pyramid

The height of a side face drawn from its vertex is called apothem. In the figure it is designated as segment ON
The point connecting the side edges and not lying in the plane of the base is called top of the pyramid(O)
Triangles that have a common side with the base and one of the vertices coinciding with the vertex are called side faces(AOD, DOC, COB, AOB)
The segment of the perpendicular drawn through the top of the pyramid to the plane of its base is called pyramid height(OK)
Diagonal section of a pyramid- this is the section passing through the top and the diagonal of the base (AOC, BOD)
A polygon that does not have a pyramid vertex is called the base of the pyramid(ABCD)

If at the base correct pyramid lies a triangle, quadrilateral, etc. then it's called regular triangular , quadrangular etc.

A triangular pyramid is a tetrahedron - a tetrahedron.

Properties of a regular pyramid

To solve problems, it is necessary to know the properties of individual elements, which are usually omitted in the condition, since it is believed that the student should know this from the very beginning.

side ribs are equal between themselves
apothems are equal
side faces are equal with each other (at the same time, respectively, their areas, sides and bases are equal), that is, they are equal triangles
all side faces are congruent isosceles triangles
in any regular pyramid, you can both inscribe and describe a sphere around it
if the centers of the inscribed and circumscribed spheres coincide, then the sum of the plane angles at the top of the pyramid is π, and each of them is π/n, respectively, where n is the number of sides of the base polygon
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
a circle can be circumscribed near the base of a regular pyramid (see also the radius of the circumscribed circle of a triangle)
all side faces form equal angles with the base plane of a regular pyramid
all heights of the side faces are equal to each other

Instructions for solving problems. The properties listed above should help in a practical solution. If you want to find the angles of inclination of the faces, their surface, etc., then general technique comes down to splitting the entire three-dimensional figure into separate flat figures and applying their properties to find individual elements of the pyramid, since many elements are common to several figures.

It is necessary to break the entire three-dimensional figure into separate elements - triangles, squares, segments. Further, to apply knowledge from the planimetry course to individual elements, which greatly simplifies finding the answer.

Formulas for the correct pyramid

Formulas for finding volume and lateral surface area:

Notation:
V - volume of the pyramid
S - base area
h - the height of the pyramid
Sb - side surface area
a - apothem (not to be confused with α)
P - base perimeter
n - number of base sides
b - side rib length
α - flat angle at the top of the pyramid

This formula for finding volume can be used only for correct pyramid:

, where

V - volume of a regular pyramid
h - the height of the regular pyramid
n is the number of sides of the regular polygon that is the base for the regular pyramid
a - side length of a regular polygon

Correct truncated pyramid

If we draw a section parallel to the base of the pyramid, then the body enclosed between these planes and the side surface is called truncated pyramid. This section for a truncated pyramid is one of its bases.

The height of the side face (which is an isosceles trapezoid) is called - apothem of a regular truncated pyramid.

A truncated pyramid is called correct if the pyramid from which it was obtained is correct.

The distance between the bases of a truncated pyramid is called truncated pyramid height
All faces of a regular truncated pyramid are isosceles (isosceles) trapezoids

Notes

See also: special cases (formulas) for a regular pyramid:

How to use the theoretical materials given here to solve your problem:

quadrangular pyramid A polyhedron is called a polyhedron whose base is a square, and all side faces are identical isosceles triangles.

This polyhedron has many different properties:

Its lateral ribs and adjacent dihedral angles are equal to each other;
The areas of the side faces are the same;
At the base of a regular quadrangular pyramid lies a square;
The height dropped from the top of the pyramid intersects with the point of intersection of the diagonals of the base.

All these properties make it easy to find . However, quite often, in addition to it, it is required to calculate the volume of the polyhedron. To do this, apply the formula for the volume of a quadrangular pyramid:

That is, the volume of the pyramid is equal to one third of the product of the height of the pyramid and the area of \u200b\u200bthe base. Since it is equal to the product of its equal sides, then we immediately enter the square area formula into the volume expression.
Consider an example of calculating the volume of a quadrangular pyramid.

Let a quadrangular pyramid be given, at the base of which lies a square with side a = 6 cm. The side face of the pyramid is b = 8 cm. Find the volume of the pyramid.

To find the volume of a given polyhedron, we need the length of its height. Therefore, we will find it by applying the Pythagorean theorem. First, let's calculate the length of the diagonal. In the blue triangle, it will be the hypotenuse. It is also worth remembering that the diagonals of the square are equal to each other and are divided in half at the intersection point:

Now from the red triangle we find the height we need h. It will be equal to:

Substitute the required values and find the height of the pyramid:

Now, knowing the height, we can substitute all the values in the formula for the volume of the pyramid and calculate the required value:

This is how, knowing a few simple formulas, we were able to calculate the volume of a regular quadrangular pyramid. Do not forget that given value measured in cubic units.

Your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please read our privacy policy and let us know if you have any questions.

Collection and use of personal information

Personal information refers to data that can be used to identify or contact a specific person.

You may be asked to provide your personal information at any time when you contact us.

The following are some examples of the types of personal information we may collect and how we may use such information.

What personal information we collect:

When you submit an application on the site, we may collect various information, including your name, phone number, email address, etc.

How we use your personal information:

Collected by us personal information allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
From time to time, we may use your personal information to send you important notices and communications.
We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
If you enter a prize draw, contest or similar incentive, we may use the information you provide to administer such programs.

Disclosure to third parties

We do not disclose information received from you to third parties.

Exceptions:

In the event that it is necessary - in accordance with the law, judicial order, in legal proceedings, and / or based on public requests or requests from state bodies in the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public interest purposes.
In the event of a reorganization, merger or sale, we may transfer the personal information we collect to the relevant third party successor.

Protection of personal information

We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as from unauthorized access, disclosure, alteration and destruction.

Maintaining your privacy at the company level

To ensure that your personal information is secure, we communicate privacy and security practices to our employees and strictly enforce privacy practices.