Definition. Side edge- this is a triangle in which one angle lies at the top of the pyramid, and the opposite side coincides with the side of the base (polygon).
Definition. Side ribs- these are the common sides of the side faces. A pyramid has as many edges as the angles of a polygon.
Definition. Pyramid height- this is a perpendicular lowered from the top to the base of the pyramid.
Definition. Apothem- this is a perpendicular to the side face of the pyramid, lowered from the top of the pyramid to the side of the base.
Definition. Diagonal section- this is a section of a pyramid by a plane passing through the top of the pyramid and the diagonal of the base.
Definition. Correct pyramid is a pyramid in which the base is a regular polygon, and the height descends to the center of the base.
Volume and surface area of the pyramid
Formula. Volume of the pyramid through base area and height:
Properties of the pyramid
If all the side edges are equal, then a circle can be drawn around the base of the pyramid, and the center of the base coincides with the center of the circle. Also, a perpendicular dropped from the top passes through the center of the base (circle).
If all the side edges are equal, then they are inclined to the plane of the base at the same angles.
The lateral ribs are equal when they form with the plane of the base equal angles or if a circle can be described around the base of the pyramid.
If side faces are inclined to the plane of the base at one angle, then a circle can be inscribed at the base of the pyramid, and the top of the pyramid is projected at its center.
If the side faces are inclined to the plane of the base at the same angle, then the apothems of the side faces are equal.
Properties of a regular pyramid
1. The top of the pyramid is equidistant from all corners of the base.
2. All side edges are equal.
3. All side ribs are inclined at equal angles to the base.
4. The apothems of all lateral faces are equal.
5. The areas of all side faces are equal.
6. All faces have the same dihedral (flat) angles.
7. A sphere can be described around the pyramid. The center of the circumscribed sphere will be the intersection point of the perpendiculars that pass through the middle of the edges.
8. You can fit a sphere into a pyramid. The center of the inscribed sphere will be the point of intersection of the bisectors emanating from the angle between the edge and the base.
9. If the center of the inscribed sphere coincides with the center of the circumscribed sphere, then the sum of the plane angles at the vertex is equal to π or vice versa, one angle is equal to π/n, where n is the number of angles at the base of the pyramid.
The connection between the pyramid and the sphere
A sphere can be described around a pyramid when at the base of the pyramid there is a polyhedron around which a circle can be described (a necessary and sufficient condition). The center of the sphere will be the intersection point of planes passing perpendicularly through the midpoints of the side edges of the pyramid.
A sphere can always be described around any triangular or regular pyramid.
A sphere can be inscribed into a pyramid if the bisector planes of the internal dihedral angles of the pyramid intersect at one point (a necessary and sufficient condition). This point will be the center of the sphere.
Connection of a pyramid with a cone
A cone is said to be inscribed in a pyramid if their vertices coincide and the base of the cone is inscribed in the base of the pyramid.
A cone can be inscribed in a pyramid if the apothems of the pyramid are equal to each other.
A cone is said to be circumscribed around a pyramid if their vertices coincide and the base of the cone is circumscribed around the base of the pyramid.
A cone can be described around a pyramid if all the lateral edges of the pyramid are equal to each other.
Relationship between a pyramid and a cylinder
A pyramid is called inscribed in a cylinder if the top of the pyramid lies on one base of the cylinder, and the base of the pyramid is inscribed in another base of the cylinder.
A cylinder can be described around a pyramid if a circle can be described around the base of the pyramid.
A tetrahedron has four faces and four vertices and six edges, where any two edges do not have common vertices but do not touch.
Each vertex consists of three faces and edges that form triangular angle.
The segment connecting the vertex of a tetrahedron with the center of the opposite face is called median of the tetrahedron(GM).
Bimedian called a segment connecting the midpoints of opposite edges that do not touch (KL).
All bimedians and medians of a tetrahedron intersect at one point (S). In this case, the bimedians are divided in half, and the medians are divided in a ratio of 3:1 starting from the top.
Definition. Acute angled pyramid- a pyramid in which the apothem is more than half the length of the side of the base.
Definition. Obtuse pyramid- a pyramid in which the apothem is less than half the length of the side of the base.
Definition. Regular tetrahedron- a tetrahedron in which all four faces are equilateral triangles. It is one of the five regular polygons. In a regular tetrahedron, all dihedral angles (between faces) and trihedral angles (at the vertex) are equal.
Definition. Rectangular tetrahedron is a tetrahedron with a right angle between three edges at the apex (the edges are perpendicular). Three faces form rectangular triangular angle and the faces are right triangles, and the base is an arbitrary triangle. The apothem of any face is equal to half the side of the base on which the apothem falls.
Definition. Isohedral tetrahedron is called a tetrahedron whose side faces are equal to each other, and the base is a regular triangle. Such a tetrahedron has faces that are isosceles triangles.
Definition. Orthocentric tetrahedron is called a tetrahedron in which all the heights (perpendiculars) that are lowered from the top to the opposite face intersect at one point.
Definition. Star pyramid called a polyhedron whose base is a star.
We continue to consider the tasks included in the Unified State Examination in mathematics. We have already studied problems where the condition is given and it is required to find the distance between two given points or an angle.
A pyramid is a polyhedron, the base of which is a polygon, the remaining faces are triangles, and they have a common vertex.
A regular pyramid is a pyramid at the base of which lies a regular polygon, and its vertex is projected into the center of the base.
A regular quadrangular pyramid - the base is a square. The top of the pyramid is projected at the intersection point of the diagonals of the base (square).
ML - apothem
∠MLO - dihedral angle at the base of the pyramid
∠MCO - angle between the lateral edge and the plane of the base of the pyramid
In this article we will look at problems to solve a regular pyramid. You need to find some element, lateral surface area, volume, height. Of course, you need to know the Pythagorean theorem, the formula for the area of the lateral surface of a pyramid, and the formula for finding the volume of a pyramid.
In the article "" presents the formulas that are necessary to solve problems in stereometry. So, the tasks:
SABCD dot O- center of the base,S vertex, SO = 51, A.C.= 136. Find side rib S.C..
IN in this case the base is a square. This means that the diagonals AC and BD are equal, they intersect and are bisected by the intersection point. Note that in a regular pyramid the height dropped from its top passes through the center of the base of the pyramid. So SO is the height and the triangleSOCrectangular. Then according to the Pythagorean theorem:
How to extract the root from large number.
Answer: 85
Decide for yourself:
In the right quadrangular pyramid SABCD dot O- center of the base, S vertex, SO = 4, A.C.= 6. Find the side edge S.C..
In a regular quadrangular pyramid SABCD dot O- center of the base, S vertex, S.C. = 5, A.C.= 6. Find the length of the segment SO.
In a regular quadrangular pyramid SABCD dot O- center of the base, S vertex, SO = 4, S.C.= 5. Find the length of the segment A.C..
SABC R- middle of the rib B.C., S- top. It is known that AB= 7, a S.R.= 16. Find the lateral surface area.
The area of the lateral surface of a regular triangular pyramid is equal to half the product of the perimeter of the base and the apothem (apothem is the height of the lateral face of a regular pyramid drawn from its vertex):
Or we can say this: the area of the lateral surface of the pyramid is equal to the sum three squares side edges. Side edges in the correct triangular pyramid are triangles of equal area. In this case:
Answer: 168
Decide for yourself:
In a regular triangular pyramid SABC R- middle of the rib B.C., S- top. It is known that AB= 1, a S.R.= 2. Find the lateral surface area.
In a regular triangular pyramid SABC R- middle of the rib B.C., S- top. It is known that AB= 1, and the area of the lateral surface is 3. Find the length of the segment S.R..
In a regular triangular pyramid SABC L- middle of the rib B.C., S- top. It is known that SL= 2, and the area of the lateral surface is 3. Find the length of the segment AB.
In a regular triangular pyramid SABC M. Area of a triangle ABC is 25, the volume of the pyramid is 100. Find the length of the segment MS.
The base of the pyramid is an equilateral triangle. That's why Mis the center of the base, andMS- height of a regular pyramidSABC. Volume of the pyramid SABC equals: view solution
In a regular triangular pyramid SABC the medians of the base intersect at the point M. Area of a triangle ABC equals 3, MS= 1. Find the volume of the pyramid.
In a regular triangular pyramid SABC the medians of the base intersect at the point M. The volume of the pyramid is 1, MS= 1. Find the area of the triangle ABC.
Let's finish here. As you can see, problems are solved in one or two steps. In the future, we will consider other problems from this part, where bodies of revolution are given, don’t miss it!
I wish you success!
Sincerely, Alexander Krutitskikh.
P.S: I would be grateful if you tell me about the site on social networks.
A triangular pyramid is a pyramid that has a triangle at its base. The height of this pyramid is the perpendicular that is lowered from the top of the pyramid to its base.
Finding the height of a pyramid
How to find the height of a pyramid? Very simple! To find the height of any triangular pyramid, you can use the volume formula: V = (1/3)Sh, where S is the area of the base, V is the volume of the pyramid, h is its height. From this formula, derive the height formula: to find the height of a triangular pyramid, you need to multiply the volume of the pyramid by 3, and then divide the resulting value by the area of the base, it will be: h = (3V)/S. Since the base of a triangular pyramid is a triangle, you can use the formula to calculate the area of a triangle. If we know: the area of the triangle S and its side z, then according to the area formula S=(1/2)γh: h = (2S)/γ, where h is the height of the pyramid, γ is the edge of the triangle; the angle between the sides of the triangle and the two sides themselves, then using the following formula: S = (1/2)γφsinQ, where γ, φ are the sides of the triangle, we find the area of the triangle. The value of the sine of angle Q needs to be looked at in the table of sines, which is available on the Internet. Next, we substitute the area value into the height formula: h = (2S)/γ. If the task requires calculating the height of a triangular pyramid, then the volume of the pyramid is already known.
Regular triangular pyramid
Find the height of a regular triangular pyramid, that is, a pyramid in which all faces are equilateral triangles, knowing the edge size γ. In this case, the edges of the pyramid are the sides of equilateral triangles. The height of a regular triangular pyramid will be: h = γ√(2/3), where γ is an edge equilateral triangle, h is the height of the pyramid. If the area of the base (S) is unknown, and only the length of the edge (γ) and the volume (V) of the polyhedron are given, then the necessary variable in the formula from the previous step must be replaced by its equivalent, which is expressed in terms of the length of the edge. The area of a triangle (regular) is equal to 1/4 of the product of the side length of this triangle squared by the square root of 3. We substitute this formula instead of the area of the base in the previous formula, and we obtain the following formula: h = 3V4/(γ 2 √3) = 12V/(γ 2 √3). The volume of a tetrahedron can be expressed through the length of its edge, then from the formula for calculating the height of the figure you can remove all variables and leave only the side triangular face figures. The volume of such a pyramid can be calculated by dividing by 12 from the product the cubed length of its face by the square root of 2.
Substituting this expression into the previous formula, we obtain the following formula for calculation: h = 12(γ 3 √2/12)/(γ 2 √3) = (γ 3 √2)/(γ 2 √3) = γ√(2 /3) = (1/3)γ√6. Also correct triangular prism can be inscribed in a sphere, and knowing only the radius of the sphere (R) one can find the height of the tetrahedron itself. The length of the edge of the tetrahedron is: γ = 4R/√6. We replace the variable γ with this expression in the previous formula and get the formula: h = (1/3)√6(4R)/√6 = (4R)/3. The same formula can be obtained by knowing the radius (R) of a circle inscribed in a tetrahedron. In this case, the length of the edge of the triangle will be equal to 12 ratios between square root of 6 and radius. We substitute this expression into the previous formula and we have: h = (1/3)γ√6 = (1/3)√6(12R)/√6 = 4R.
How to find the height of a regular quadrangular pyramid
To answer the question of how to find the length of the height of a pyramid, you need to know what a regular pyramid is. A quadrangular pyramid is a pyramid that has a quadrangle at its base. If in the conditions of the problem we have: the volume (V) and the area of the base (S) of the pyramid, then the formula for calculating the height of the polyhedron (h) will be as follows - divide the volume multiplied by 3 by the area S: h = (3V)/S. Given a square base of a pyramid with a given volume (V) and side length γ, replace the area (S) in the previous formula with the square of the side length: S = γ 2 ; H = 3V/γ2. The height of a regular pyramid h = SO passes exactly through the center of the circle that is circumscribed near the base. Since the base of this pyramid is a square, point O is the intersection point of diagonals AD and BC. We have: OC = (1/2)BC = (1/2)AB√6. Next, we are in right triangle We find SOC (using the Pythagorean theorem): SO = √(SC 2 -OC 2). Now you know how to find the height of a regular pyramid.
Introduction
When we started studying stereometric figures, we touched on the topic “Pyramid”. We liked this topic because the pyramid is very often used in architecture. And since ours future profession architect, inspired by this figure, we think that she can push us to great projects.
The strength of architectural structures is their most important quality. Linking strength, firstly, with the materials from which they are created, and, secondly, with the features of design solutions, it turns out that the strength of a structure is directly related to the geometric shape that is basic for it.
In other words, we are talking about a geometric figure that can be considered as a model of the corresponding architectural form. It turns out that geometric shape also determines the strength of an architectural structure.
Since ancient times, the Egyptian pyramids have been considered the most durable architectural structures. As you know, they have the shape of regular quadrangular pyramids.
It is this geometric shape that provides the greatest stability due to the large base area. On the other hand, the pyramid shape ensures that the mass decreases as the height above the ground increases. It is these two properties that make the pyramid stable, and therefore strong under the conditions of gravity.
Objective of the project: learn something new about pyramids, deepen your knowledge and find practical application.
To achieve this goal, it was necessary to solve the following tasks:
· Learn historical information about the pyramid
· Consider the pyramid as geometric figure
· Find application in life and architecture
· Find the similarities and differences between the pyramids located in different parts Sveta
Theoretical part
Historical information
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 the volume of the pyramid was Democritus, and Eudoxus of Cnidus proved it. The ancient Greek mathematician Euclid systematized knowledge about the pyramid in the XII volume of his “Elements”, and also derived the first definition of a pyramid: a solid figure bounded by planes that converge from one plane to one point.
Tombs of Egyptian pharaohs. The largest of them - the pyramids of Cheops, Khafre and Mikerin in El Giza - were considered one of the Seven Wonders of the World in ancient times. The construction of the pyramid, in which the Greeks and Romans already saw a monument to the unprecedented pride of kings and cruelty that doomed the entire people of Egypt to meaningless construction, was the most important cult act and was supposed to express, apparently, the mystical identity of the country and its ruler. The population of the country worked on the construction of the tomb during the part of the year free from agricultural work. A number of texts testify to the attention and care that the kings themselves (albeit of a later time) paid to the construction of their tomb and its builders. It is also known about the special cult honors that were given to the pyramid itself.
Basic Concepts
Pyramid is a polyhedron whose base is a polygon, and the remaining faces are triangles that have a common vertex.
Apothem- the height of the side face of a regular pyramid, drawn from its vertex;
Side faces- triangles meeting at a vertex;
Side ribs- common sides of the side faces;
Top of the pyramid- a point connecting the side ribs and not lying in the plane of the base;
Height- a perpendicular segment 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- section of the pyramid passing through the top and diagonal of the base;
Base- a polygon that does not belong to the vertex of the pyramid.
Basic properties of a regular pyramid
The lateral edges, lateral faces and apothems are respectively equal.
The dihedral angles at the base are equal.
The dihedral angles at the lateral edges are equal.
Each height point is equidistant from all the vertices of the base.
Each height point is equidistant from all side faces.
Basic pyramid formulas
Side area and full surface pyramids.
The area of the lateral surface of a pyramid (full and truncated) is the sum of the areas of all its lateral faces, the total surface area is the sum of the areas of all its faces.
Theorem: 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 of the pyramid.
p- base perimeter;
h- apothem.
The area of the lateral and full surfaces of a truncated pyramid.
p 1, p 2 - base perimeters;
h- apothem.
R- total surface area of a regular truncated pyramid;
S side- area of the lateral surface of a regular truncated pyramid;
S 1 + S 2- base area
Volume of the pyramid
Form volume ula is used for pyramids of any kind.
H- height of the pyramid.
Pyramid corners
The angles formed by the side face and the base of the pyramid are called dihedral angles at the base of the pyramid.
A dihedral angle is formed by two perpendiculars.
To determine this angle, you often need to use the three perpendicular theorem.
The angles formed by the lateral edge and its projection onto the base plane are called angles between the side edge and the plane of the base.
The angle formed by two lateral edges is called dihedral angle at the lateral edge of the pyramid.
The angle formed by two lateral edges of one face of the pyramid is called angle at the top of the pyramid.
Pyramid sections
The surface of a pyramid is the surface of a polyhedron. Each of its faces is a plane, so the section of a pyramid defined by a cutting plane is broken line, consisting of individual straight lines.
Diagonal section
The section of a pyramid by a plane passing through two lateral edges that do not lie on the same face is called diagonal section pyramids.
Parallel sections
Theorem:
If the pyramid is intersected by a plane parallel to the base, then the lateral edges and heights of the pyramid are divided by this plane into proportional parts;
The section of this plane is a polygon similar to the base;
The areas of the section and the base are related to each other as the squares of their distances from the vertex.
Types of pyramid
Correct pyramid– a pyramid whose base is a regular polygon, and the top of the pyramid is projected into the center of the base.
For a regular pyramid:
1. side ribs are equal
2. side faces are equal
3. apothems are equal
4. dihedral angles at the base are equal
5. dihedral angles at the lateral edges are equal
6. each point of height is equidistant from all vertices of the base
7. each height point is equidistant from all side edges
Truncated pyramid- part of the pyramid enclosed between its base and a cutting plane parallel to the base.
The base and corresponding section of a truncated pyramid are called bases of a truncated pyramid.
A perpendicular drawn from any point of one base to the plane of another is called the height of a truncated pyramid.
Tasks
No. 1. In a regular quadrangular pyramid, point O is the center of the base, SO=8 cm, BD=30 cm. Find the side edge SA.
Problem solving
No. 1. In a regular pyramid, all faces and edges are equal.
Consider OSB: OSB is a rectangular rectangle, because.
SB 2 =SO 2 +OB 2
SB 2 =64+225=289
Pyramid in architecture
A pyramid is a monumental structure in the shape of an ordinary regular geometric pyramid, wherein sides converge at one point. According to their functional purpose, pyramids in ancient times were places of burial or cult worship. The base of a pyramid can be triangular, quadrangular, or in the shape of a polygon with an arbitrary number of vertices, but the most common version is the quadrangular base.
There are a considerable number of pyramids built different cultures Ancient world mainly as temples or monuments. Large pyramids include the Egyptian pyramids.
All over the Earth you can see architectural structures in the form of pyramids. The pyramid buildings are reminiscent of ancient times and look very beautiful.
Egyptian pyramids greatest architectural monuments Ancient Egypt, among which one of the “Seven Wonders of the World” is the Pyramid of Cheops. From the foot to the top it reaches 137.3 m, and before it lost the top, its height was 146.7 m
The radio station building in the capital of Slovakia, resembling an inverted pyramid, was built in 1983. In addition to offices and service premises, inside the volume there is a fairly spacious concert hall, which has one of the largest organs in Slovakia.
The Louvre, which “is silent and majestic, like a pyramid,” has undergone many changes over the centuries before becoming greatest museum peace. It was born as a fortress, erected by Philip Augustus in 1190, which soon became a royal residence. In 1793 the palace became a museum. Collections are enriched through bequests or purchases.
