This video tutorial will help users get an idea of the Pyramid theme. Correct pyramid. In this lesson we will get acquainted with the concept of a pyramid and give it a definition. Let's consider what a regular pyramid is and what properties it has. Then we prove the theorem about the lateral surface of a regular pyramid.
In this lesson we will get acquainted with the concept of a pyramid and give it a definition.
Consider a polygon A 1 A 2...A n, which lies in the α plane, and the point P, which does not lie in the α plane (Fig. 1). Let's connect the dots P with peaks A 1, A 2, A 3, … A n. We get n triangles: A 1 A 2 R, A 2 A 3 R and so on.
Definition. Polyhedron RA 1 A 2 ...A n, made up of n-square A 1 A 2...A n And n triangles RA 1 A 2, RA 2 A 3 …RA n A n-1 is called n-coal pyramid. Rice. 1.
Rice. 1
Consider a quadrangular pyramid PABCD(Fig. 2).
R- the top of the pyramid.
ABCD- the base of the pyramid.
RA- side rib.
AB- base rib.
From point R let's drop the perpendicular RN to the base plane ABCD. The perpendicular drawn is the height of the pyramid.
Rice. 2
The full surface of the pyramid consists of the lateral surface, that is, the area of all lateral faces, and the area of the base:
S full = S side + S main
A pyramid is called correct if:
- its base is a regular polygon;
- the segment connecting the top of the pyramid to the center of the base is its height.
Explanation using an example of the correct quadrangular pyramid
Consider a regular quadrangular pyramid PABCD(Fig. 3).
R- the top of the pyramid. Base of the pyramid ABCD- a regular quadrilateral, that is, a square. Dot ABOUT, the point of intersection of the diagonals, is the center of the square. Means, RO is the height of the pyramid.
Rice. 3
Explanation: in the correct n In a triangle, the center of the inscribed circle and the center of the circumcircle coincide. This center is called the center of the polygon. Sometimes they say that the vertex is projected into the center.
The height of the lateral face of a regular pyramid drawn from its vertex is called apothem and is designated h a.
1. everything lateral ribs of a regular pyramid are equal;
2. side faces are congruent isosceles triangles.
We will give a proof of these properties using the example of a regular quadrangular pyramid.
Given: PABCD- regular quadrangular pyramid,
ABCD- square,
RO- height of the pyramid.
Prove:
1. RA = PB = RS = PD
2.∆ABP = ∆BCP =∆CDP =∆DAP See Fig. 4.
Rice. 4
Proof.
RO- height of the pyramid. That is, straight RO perpendicular to the plane ABC, and therefore direct JSC, VO, SO And DO lying in it. So triangles ROA, ROV, ROS, ROD- rectangular.
Consider a square ABCD. From the properties of a square it follows that AO = VO = CO = DO.
Then the right triangles ROA, ROV, ROS, ROD leg RO- general and legs JSC, VO, SO And DO are equal, which means that these triangles are equal on two sides. From the equality of triangles follows the equality of segments, RA = PB = RS = PD. Point 1 has been proven.
Segments AB And Sun are equal because they are sides of the same square, RA = PB = RS. So triangles AVR And VSR - isosceles and equal on three sides.
In a similar way we find that triangles ABP, VCP, CDP, DAP are isosceles and equal, as required to be proved in paragraph 2.
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:
To prove this, let’s choose a regular triangular pyramid.
Given: RAVS- regular triangular pyramid.
AB = BC = AC.
RO- height.
Prove: 
. See Fig. 5.
Rice. 5
Proof.
RAVS- regular triangular pyramid. That is AB= AC = BC. Let ABOUT- center of the triangle ABC, Then RO is the height of the pyramid. At the base of the pyramid lies equilateral triangle ABC. Note that 
.
Triangles RAV, RVS, RSA- equal isosceles triangles (by property). U triangular pyramid three side faces: RAV, RVS, RSA. This means that the area of the lateral surface of the pyramid is:
S side = 3S RAW
The theorem has been proven.
The radius of a circle inscribed at the base of a regular quadrangular pyramid is 3 m, the height of the pyramid is 4 m. Find the area of the lateral surface of the pyramid.
Given: regular quadrangular pyramid ABCD,
ABCD- square,
r= 3 m,
RO- height of the pyramid,
RO= 4 m.
Find: S side. See Fig. 6.
Rice. 6
Solution.
According to the proven theorem, .
Let's first find the side of the base AB. We know that the radius of a circle inscribed at the base of a regular quadrangular pyramid is 3 m.
Then, m.
Find the perimeter of the square ABCD with a side of 6 m:
Consider a triangle BCD. Let M- middle of the side DC. Because ABOUT- middle BD, That 
(m).
Triangle DPC- isosceles. M- middle DC. That is, RM- median, and therefore height in the triangle DPC. Then RM- apothem of the pyramid.
RO- height of the pyramid. Then, straight RO perpendicular to the plane ABC, and therefore direct OM, lying in it. Let's find the apothem RM from a right triangle ROM.
Now we can find lateral surface pyramids:
Answer: 60 m2.
The radius of the circle circumscribed around the base of a regular triangular pyramid is equal to m. The area of the lateral surface is 18 m 2. Find the length of the apothem.
Given: ABCP- regular triangular pyramid,
AB = BC = SA,
R= m,
S side = 18 m2.
Find: . See Fig. 7.
Rice. 7
Solution.
In a right triangle ABC The radius of the circumscribed circle is given. Let's find a side AB this triangle using the law of sines.
Knowing the side of a regular triangle (m), we find its perimeter.
By the theorem on the lateral surface area of a regular pyramid, where h a- apothem of the pyramid. Then:
Answer: 4 m.
So, we looked at what a pyramid is, what a regular pyramid is, and we proved the theorem about the lateral surface of a regular pyramid. In the next lesson we will get acquainted with the truncated pyramid.
References
- Geometry. Grades 10-11: textbook for students of general education institutions (basic and specialized levels) / I. M. Smirnova, V. A. Smirnov. - 5th ed., rev. and additional - M.: Mnemosyne, 2008. - 288 p.: ill.
- Geometry. 10-11 grade: Textbook for general education educational institutions/ Sharygin I.F. - M.: Bustard, 1999. - 208 p.: ill.
- Geometry. Grade 10: Textbook for general education institutions with in-depth and specialized study of mathematics /E. V. Potoskuev, L. I. Zvalich. - 6th ed., stereotype. - M.: Bustard, 008. - 233 p.: ill.
- Internet portal "Yaklass" ()
- Internet portal “Festival of pedagogical ideas “First of September” ()
- Internet portal “Slideshare.net” ()
Homework
- Can a regular polygon be the base of an irregular pyramid?
- Prove that disjoint edges of a regular pyramid are perpendicular.
- Find the value of the dihedral angle at the side of the base of a regular quadrangular pyramid if the apothem of the pyramid is equal to the side of its base.
- RAVS- regular triangular pyramid. Construct the linear angle of the dihedral angle at the base of the pyramid.
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 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 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 in 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 edges 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 called a tetrahedron whose side faces are equal to each other and the base is 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 A polyhedron whose base is a star is called.
Pyramid. Truncated pyramid
Pyramid is a polyhedron, one of whose faces is a polygon ( base ), and all other faces are triangles with a common vertex ( side faces ) (Fig. 15). The pyramid is called correct , if its base is a regular polygon and the top of the pyramid is projected into the center of the base (Fig. 16). A triangular pyramid with all edges equal is called tetrahedron .
Lateral rib of a pyramid is the side of the side face that does not belong to the base Height pyramid is the distance from its top to the plane of the base. All lateral edges of a regular pyramid are equal to each other, all lateral faces are equal isosceles triangles. The height of the side face of a regular pyramid drawn from the vertex is called apothem . Diagonal section is called a section of a pyramid by a plane passing through two lateral edges that do not belong to the same face.
Lateral surface area pyramid is the sum of the areas of all lateral faces. Area full surface is called the sum of the areas of all the side faces and the base.
Theorems
1. If in a pyramid all the lateral edges are equally inclined to the plane of the base, then the top of the pyramid is projected into the center of the circle circumscribed near the base.
2. If in a pyramid all lateral edges have equal lengths, then the top of the pyramid is projected into the center of the circle circumscribed near the base.
3. If all the faces in a pyramid are equally inclined to the plane of the base, then the top of the pyramid is projected into the center of a circle inscribed in the base.
To calculate the volume of an arbitrary pyramid, the correct formula is:
Where V- volume;
S base– base area;
H– height of the pyramid.
For a regular pyramid, the following formulas are correct:
Where p– base perimeter;
h a– apothem;
H- height;
S full
S side
S base– base area;
V– volume of a regular pyramid.
Truncated pyramid called the part of the pyramid enclosed between the base and a cutting plane parallel to the base of the pyramid (Fig. 17). Regular truncated pyramid is the part of a regular pyramid enclosed between the base and a cutting plane parallel to the base of the pyramid.
Reasons truncated pyramid - similar polygons. Side faces – trapezoids. Height of a truncated pyramid is the distance between its bases. Diagonal a truncated pyramid is a segment connecting its vertices that do not lie on the same face. Diagonal section is a section of a truncated pyramid by a plane passing through two lateral edges that do not belong to the same face.
For a truncated pyramid the following formulas are valid:
(4)
Where S 1 , S 2 – areas of the upper and lower bases;
S full– total surface area;
S side– lateral surface area;
H- height;
V– volume of a truncated pyramid.
For a regular truncated pyramid the formula is correct:
Where p 1 , p 2 – perimeters of the bases;
h a– apothem of a regular truncated pyramid.
Example 1. In a regular triangular pyramid, the dihedral angle at the base is 60º. Find the tangent of the angle of inclination of the side edge to the plane of the base.
Solution. Let's make a drawing (Fig. 18).
 |
The pyramid is regular, which means that at the base there is an equilateral triangle and all the side faces are equal isosceles triangles. The dihedral angle at the base is the angle of inclination of the side face of the pyramid to the plane of the base. The linear angle is the angle a between two perpendiculars: etc. The top of the pyramid is projected at the center of the triangle (the center of the circumcircle and inscribed circle of the triangle ABC). The angle of inclination of the side edge (for example S.B.) is the angle between the edge itself and its projection onto the plane of the base. For the rib S.B. this angle will be the angle SBD. To find the tangent you need to know the legs SO And O.B.. Let the length of the segment BD equals 3 A. Dot ABOUT segment BD is divided into parts: and From we find SO: 
From we find: 
Answer:
Example 2. Find the volume of a regular truncated quadrangular pyramid if the diagonals of its bases are equal to cm and cm, and its height is 4 cm.
Solution. To find the volume of a truncated pyramid, we use formula (4). To find the area of the bases, you need to find the sides of the base squares, knowing their diagonals. The sides of the bases are equal to 2 cm and 8 cm, respectively. This means the areas of the bases and Substituting all the data into the formula, we calculate the volume of the truncated pyramid:
Answer: 112 cm 3.
Example 3. Find the area of the lateral face of a regular triangular truncated pyramid, the sides of the bases of which are 10 cm and 4 cm, and the height of the pyramid is 2 cm.
Solution. Let's make a drawing (Fig. 19).
The side face of this pyramid is an isosceles trapezoid. To calculate the area of a trapezoid, you need to know the base and height. The bases are given according to the condition, only the height remains unknown. We'll find her from where A 1 E perpendicular from a point A 1 on the plane of the lower base, A 1 D– perpendicular from A 1 per AC. A 1 E= 2 cm, since this is the height of the pyramid. To find DE Let's make an additional drawing showing the top view (Fig. 20). Dot ABOUT– projection of the centers of the upper and lower bases. since (see Fig. 20) and On the other hand OK– radius inscribed in the circle and 
OM– radius inscribed in a circle:
MK = DE.
According to the Pythagorean theorem from
Side face area: 
Answer:
Example 4. At the base of the pyramid lies an isosceles trapezoid, the bases of which A And b (a> b). Each side face forms an angle equal to the plane of the base of the pyramid j. Find the total surface area of the pyramid.
Solution. Let's make a drawing (Fig. 21). Total surface area of the pyramid SABCD equal to the sum of the areas and the area of the trapezoid ABCD.
Let us use the statement that if all the faces of the pyramid are equally inclined to the plane of the base, then the vertex is projected into the center of the circle inscribed in the base. Dot ABOUT– vertex projection S at the base of the pyramid. Triangle SOD is the orthogonal projection of the triangle CSD to the plane of the base. By the theorem on the area of orthogonal projection flat figure we get:
Likewise it means 
Thus, the problem was reduced to finding the area of the trapezoid ABCD. Let's draw a trapezoid ABCD separately (Fig. 22). Dot ABOUT– the center of a circle inscribed in a trapezoid.
Since a circle can be inscribed in a trapezoid, then or From the Pythagorean theorem we have
Here you can find basic information about pyramids and related formulas and concepts. All of them are studied with a mathematics tutor in preparation for the Unified State Exam.
Consider a plane, a polygon 
, lying in it and a point S, not lying in it. Let's connect S to all the vertices of the polygon. The resulting polyhedron is called a pyramid. The segments are called side ribs. 
The polygon is called the base, and point S is 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. An alternative name for a triangular pyramid is tetrahedron. The height of a pyramid is the perpendicular descending from its top to the plane of the base. 
A pyramid is called regular if a regular polygon, and the base of the pyramid's altitude (the base of the perpendicular) is its center.
Tutor's comment:
Do not confuse the concepts 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 are equal. This is his definition. It is easy to prove that the equality implies that the center P of the polygon coincides with a base height, 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. 
A mathematics tutor about his terminology: 80% of work with pyramids is 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 call the first of them apothemal, and the 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 area of the total surface of the pyramid.
3) 
, where MN is the distance between any two crossing edges, and is the area of the parallelogram formed by the midpoints of the four remaining edges.
Property of the base of the height of a pyramid:
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 to 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 comment: Please note that all points are united by one common property: one way or another, lateral faces are involved everywhere (apothems are their elements). Therefore, the tutor can offer a less precise, but more convenient for learning, formulation: 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 is enough to show that all apothem triangles are equal.
Point P coincides with the center of a circle circumscribed near the base of the pyramid if one of three conditions is true:
1) All side edges are equal
2) All side ribs are equally inclined to the base
3) All side ribs are equally inclined to the height