The pyramid has its base, side ribs and height. Pyramid. Visual Guide (2019)
- apothem- the height of the side face of a regular pyramid, which is drawn from its vertex (in addition, the apothem is the length of the perpendicular, which is lowered from the middle of the regular polygon to one of its sides);
- side faces (ASB, BSC, CSD, DSA) - triangles that meet at the vertex;
- lateral ribs ( AS , B.S. , C.S. , D.S. ) — common sides of the side faces;
- top of the pyramid (t. S) - a point that connects the side ribs and which does not lie in the plane of the base;
- height ( SO ) - a perpendicular segment 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 the pyramid- a section of the pyramid that passes through the top and the diagonal of the base;
- base (ABCD) - a polygon that does not belong to the vertex of the pyramid.
Properties of the pyramid.
1. When all side edges are of the same size, then:
- it is easy to describe a circle near the base of the pyramid, and the top of the pyramid will be projected into the center of this circle;
- the side ribs form equal angles with the plane of the base;
- Moreover, the opposite is also true, i.e. when the side ribs form equal angles with the plane of the base, or when a circle can be described around the base of the pyramid and the top of the pyramid will be projected into the center of this circle, it means that all the side edges of the pyramid are the same size.
2. When the side faces have an angle of inclination to the plane of the base of the same value, then:
- it is easy to describe a circle near the base of the pyramid, and 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 equal to ½ the product of the perimeter of the base and the height of the side face.
3. A sphere can be described around a pyramid if at the base of the pyramid there 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 middles 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 into 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.
Based on the number of angles, the base of the pyramid is divided into triangular, quadrangular, and so on.
There will be a pyramid triangular, quadrangular, and so on, when the base of the pyramid is a triangle, a quadrangle, and so on. A triangular pyramid is a tetrahedron - a tetrahedron. Quadrangular - pentagonal and so on.
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 towards 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 constructive 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.
Project goal: 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 called 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 plane of the base 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, therefore the section of a pyramid defined by a cutting plane is a 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 correct 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 form of an ordinary regular geometric pyramid, in which sides converge at one point. By 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 by 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, immutable and majestic, like a pyramid,” has undergone many changes over the centuries before becoming the greatest museum in the world. It was born as a fortress, erected by Philip Augustus in 1190, which soon became royal residence. In 1793 the palace became a museum. Collections are enriched through bequests or purchases.
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 falls 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 edges are equal when they form equal angles with the plane of the base or if a circle can be described around the base of the pyramid.
If the side faces are inclined to the plane of the base at the same angle, then a circle can be inscribed into the base of the pyramid, and the top of the pyramid is projected into 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.
It is always possible to describe a sphere 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 called a tetrahedron in which there is 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.
Entry level
Pyramid. Visual guide (2019)
What is a pyramid?
What does she look like?
You see: at the bottom of the pyramid (they say “ at the base") some polygon, and all the vertices of this polygon are connected to some point in space (this point is called " vertex»).
This whole structure still has side faces, side ribs And base ribs. Once again, let's draw a pyramid along with all these names:
Some pyramids may look very strange, but they are still pyramids.
Here, for example, is completely “oblique” pyramid.
And a little more about the names: if there is a triangle at the base of the pyramid, then the pyramid is called triangular, if it is a quadrangle, then quadrangular, and if it is a centagon, then... guess for yourself.
At the same time, the point where it fell height, called height base. Please note that in the “crooked” pyramids height may even end up outside the pyramid. Like this:
And there’s nothing wrong with that. It looks like an obtuse triangle.
Correct pyramid.
Lots of complicated words? Let's decipher: “At the base - correct” - this is understandable. Now let us remember that a regular polygon has a center - a point that is the center of and , and .
Well, the words “the top is projected into the center of the base” mean that the base of the height falls exactly into the center of the base. Look how smooth and cute it looks regular pyramid.
Hexagonal: at the base there is a regular hexagon, the vertex is projected into the center of the base.
Quadrangular: the base is a square, the top is projected to the point of intersection of the diagonals of this square.
Triangular: at the base there is a regular triangle, the vertex is projected to the point of intersection of the heights (they are also medians and bisectors) of this triangle.
Very important properties of a regular pyramid:
In the right pyramid
- all side edges are equal.
- all lateral faces are isosceles triangles and all these triangles are equal.
Volume of the pyramid
The main formula for the volume of a pyramid:
Where exactly did it come from? This is not so simple, and at first you just need to remember that a pyramid and a cone have volume in the formula, but a cylinder does not.
Now let's calculate the volume of the most popular pyramids.
Let the side of the base be equal and the side edge equal. We need to find and.
This is the area regular triangle.
Let's remember how to look for this area. We use the area formula:
For us, “ ” is this, and “ ” is also this, eh.
Now let's find it.
According to the Pythagorean theorem for
What's the difference? This is the circumradius in because pyramidcorrect and, therefore, the center.
Since - the point of intersection of the medians too.
(Pythagorean theorem for)
Let's substitute it into the formula for.
And let’s substitute everything into the volume formula:
Attention: if you have a regular tetrahedron (i.e.), then the formula turns out like this:
Let the side of the base be equal and the side edge equal.
There is no need to look here; After all, the base is a square, and therefore.
We'll find it. According to the Pythagorean theorem for
Do we know? Well, almost. Look:
(we saw this by looking at it).
Substitute into the formula for:
And now we substitute and into the volume formula.
Let the side of the base be equal and the side edge.
How to find? Look, a hexagon consists of exactly six identical regular triangles. We have already looked for the area of a regular triangle when calculating the volume of a regular triangular pyramid; here we use the formula we found.
Now let's find (it).
According to the Pythagorean theorem for
But what does it matter? It's simple because (and everyone else too) is correct.
Let's substitute:
\displaystyle V=\frac(\sqrt(3))(2)((a)^(2))\sqrt(((b)^(2))-((a)^(2)))
PYRAMID. BRIEFLY ABOUT THE MAIN THINGS
A pyramid is a polyhedron that consists of any flat polygon (), a point not lying in the plane of the base (top of the pyramid) and all segments connecting the top of the pyramid with points of the base (side edges).
A perpendicular dropped from the top of the pyramid to the plane of the base.
Correct pyramid- a pyramid in which a regular polygon lies at the base, and the top of the pyramid is projected into the center of the base.
Property of a regular pyramid:
- In a regular pyramid, all lateral edges are equal.
- All lateral faces are isosceles triangles and all these triangles are equal.
When solving Problem C2 using the coordinate method, many students face the same problem. They can't calculate coordinates of points included in the scalar product formula. The greatest difficulties arise pyramids. And if the base points are considered more or less normal, then the tops are a real hell.
Today we will work on a regular quadrangular pyramid. There are more triangular pyramid(aka - tetrahedron). It's more complex design, so a separate lesson will be devoted to it.
First, let's remember the definition:
A regular pyramid is one that:
- The base is a regular polygon: triangle, square, etc.;
- An altitude drawn to the base passes through its center.
In particular, the base of a quadrangular pyramid is square. Just like Cheops, only a little smaller.
Below are calculations for a pyramid in which all edges are equal to 1. If this is not the case in your problem, the calculations do not change - just the numbers will be different.
Vertices of a quadrangular pyramid
So, let a regular quadrangular pyramid SABCD be given, where S is the vertex and the base ABCD is a square. All edges are equal to 1. You need to enter a coordinate system and find the coordinates of all points. We have:
We introduce a coordinate system with origin at point A:
- The OX axis is directed parallel to the edge AB;
- OY axis is parallel to AD. Since ABCD is a square, AB ⊥ AD;
- Finally, we direct the OZ axis upward, perpendicular to the ABCD plane.
Now we calculate the coordinates. Additional construction: SH - height drawn to the base. For convenience, we will place the base of the pyramid in a separate drawing. Since points A, B, C and D lie in the OXY plane, their coordinate is z = 0. We have:
- A = (0; 0; 0) - coincides with the origin;
- B = (1; 0; 0) - step by 1 along the OX axis from the origin;
- C = (1; 1; 0) - step by 1 along the OX axis and by 1 along the OY axis;
- D = (0; 1; 0) - step only along the OY axis.
- H = (0.5; 0.5; 0) - the center of the square, the middle of the segment AC.
It remains to find the coordinates of point S. Note that the x and y coordinates of points S and H are the same, since they lie on a line parallel to the OZ axis. It remains to find the z coordinate for point S.
Consider triangles ASH and ABH:
- AS = AB = 1 by condition;
- Angle AHS = AHB = 90°, since SH is the height and AH ⊥ HB as the diagonals of the square;
- Side AH is common.
Hence, right triangles ASH and ABH equal one leg and one hypotenuse each. This means SH = BH = 0.5 BD. But BD is the diagonal of a square with side 1. Therefore we have:
Total coordinates of point S:
In conclusion, we write down the coordinates of all the vertices of a regular rectangular pyramid:
What to do when the ribs are different
What if the side edges of the pyramid are not equal to the edges of the base? In this case, consider the triangle AHS:
Triangle AHS - rectangular, and the hypotenuse AS is also a side edge of the original pyramid SABCD. Leg AH is easily calculated: AH = 0.5 AC. We will find the remaining leg SH according to the Pythagorean theorem. This will be the z coordinate for point S.
Task. Given a regular quadrangular pyramid SABCD, at the base of which lies a square with side 1. Side edge BS = 3. Find the coordinates of point S.
We already know the x and y coordinates of this point: x = y = 0.5. This follows from two facts:
- The projection of point S onto the OXY plane is point H;
- At the same time, point H is the center of a square ABCD, all sides of which are equal to 1.
It remains to find the coordinate of point S. Consider triangle AHS. It is rectangular, with the hypotenuse AS = BS = 3, the leg AH being half the diagonal. For further calculations we need its length:
Pythagorean theorem for triangle AHS: AH 2 + SH 2 = AS 2. We have:
So, the coordinates of point S:
