Definition
Pyramid is a polyhedron composed of a polygon \(A_1A_2...A_n\) and \(n\) triangles with a common vertex \(P\) (not lying in the plane of the polygon) and opposite sides coinciding with the sides of the polygon.
Designation: \(PA_1A_2...A_n\) .
Example: pentagonal pyramid \(PA_1A_2A_3A_4A_5\) .
Triangles \(PA_1A_2, \ PA_2A_3\) etc. called side faces pyramids, segments \(PA_1, PA_2\), etc. - side ribs, polygon \(A_1A_2A_3A_4A_5\) – basis, point \(P\) – summit.
Height Pyramids are a perpendicular dropped from the top of the pyramid to the plane of the base.
A pyramid with a triangle at its base is called tetrahedron.
The pyramid is called correct, if its base is a regular polygon and one of the following conditions is met:
\((a)\) side edges of the pyramid are equal;
\((b)\) the height of the pyramid passes through the center of the circumscribed circle near the base;
\((c)\) side ribs are inclined to the base plane at the same angle.
\((d)\) side faces are inclined to the base plane at the same angle.
regular tetrahedron is a triangular pyramid, all the faces of which are equal equilateral triangles.
Theorem
The conditions \((a), (b), (c), (d)\) are equivalent.
Proof
Draw the height of the pyramid \(PH\) . Let \(\alpha\) be the plane of the base of the pyramid.
1) Let us prove that \((a)\) implies \((b)\) . Let \(PA_1=PA_2=PA_3=...=PA_n\) .
Because \(PH\perp \alpha\) , then \(PH\) is perpendicular to any line lying in this plane, so the triangles are right-angled. So these triangles are equal in common leg \(PH\) and hypotenuse \(PA_1=PA_2=PA_3=...=PA_n\) . So \(A_1H=A_2H=...=A_nH\) . This means that the points \(A_1, A_2, ..., A_n\) are at the same distance from the point \(H\) , therefore, they lie on the same circle with radius \(A_1H\) . This circle, by definition, is circumscribed about the polygon \(A_1A_2...A_n\) .
2) Let us prove that \((b)\) implies \((c)\) .
\(PA_1H, PA_2H, PA_3H,..., PA_nH\) rectangular and equal in two legs. Hence, their angles are also equal, therefore, \(\angle PA_1H=\angle PA_2H=...=\angle PA_nH\).
3) Let us prove that \((c)\) implies \((a)\) .
Similar to the first point, triangles \(PA_1H, PA_2H, PA_3H,..., PA_nH\) rectangular and along the leg and acute angle. This means that their hypotenuses are also equal, that is, \(PA_1=PA_2=PA_3=...=PA_n\) .
4) Let us prove that \((b)\) implies \((d)\) .
Because in a regular polygon, the centers of the circumscribed and inscribed circles coincide (generally speaking, this point is called the center of a regular polygon), then \(H\) is the center of the inscribed circle. Let's draw perpendiculars from the point \(H\) to the sides of the base: \(HK_1, HK_2\), etc. These are the radii of the inscribed circle (by definition). Then, according to the TTP, (\(PH\) is a perpendicular to the plane, \(HK_1, HK_2\), etc. are projections perpendicular to the sides) oblique \(PK_1, PK_2\), etc. perpendicular to the sides \(A_1A_2, A_2A_3\), etc. respectively. So, by definition \(\angle PK_1H, \angle PK_2H\) equal to the angles between the side faces and the base. Because triangles \(PK_1H, PK_2H, ...\) are equal (as right-angled on two legs), then the angles \(\angle PK_1H, \angle PK_2H, ...\) are equal.
5) Let us prove that \((d)\) implies \((b)\) .
Similarly to the fourth point, the triangles \(PK_1H, PK_2H, ...\) are equal (as rectangular along the leg and acute angle), which means that the segments \(HK_1=HK_2=...=HK_n\) are equal. Hence, by definition, \(H\) is the center of a circle inscribed in the base. But since for regular polygons, the centers of the inscribed and circumscribed circles coincide, then \(H\) is the center of the circumscribed circle. Chtd.
Consequence
The side faces of a regular pyramid are equal isosceles triangles.
Definition
The height of the side face of a regular pyramid, drawn from its top, is called apothema.
The apothems of all lateral faces of a regular pyramid are equal to each other and are also medians and bisectors.
Important Notes
1. The height of a regular triangular pyramid falls to the intersection point of the heights (or bisectors, or medians) of the base (the base is a regular triangle).
2. The height of a regular quadrangular pyramid falls to the point of intersection of the diagonals of the base (the base is a square).
3. The height of a regular hexagonal pyramid falls to the point of intersection of the diagonals of the base (the base is a regular hexagon).
4. The height of the pyramid is perpendicular to any straight line lying at the base.
Definition
The pyramid is called rectangular if one of its lateral edges is perpendicular to the plane of the base.
Important Notes
1. For a rectangular pyramid, the edge perpendicular to the base is the height of the pyramid. That is, \(SR\) is the height.
2. Because \(SR\) perpendicular to any line from the base, then \(\triangle SRM, \triangle SRP\) are right triangles.
3. Triangles \(\triangle SRN, \triangle SRK\) are also rectangular.
That is, any triangle formed by this edge and the diagonal coming out of the vertex of this edge, which lies at the base, will be right-angled.
\[(\Large(\text(Volume and surface area of the pyramid)))\]
Theorem
The volume of a pyramid is equal to one third of the product of the area of the base and the height of the pyramid: \
Consequences
Let \(a\) be the side of the base, \(h\) be the height of the pyramid.
1. The volume of a regular triangular pyramid is \(V_(\text(right triangle pyr.))=\dfrac(\sqrt3)(12)a^2h\),
2. The volume of a regular quadrangular pyramid is \(V_(\text(right.four.pyre.))=\dfrac13a^2h\).
3. The volume of a regular hexagonal pyramid is \(V_(\text(right.hex.pyr.))=\dfrac(\sqrt3)(2)a^2h\).
4. The volume of a regular tetrahedron is \(V_(\text(right tetra.))=\dfrac(\sqrt3)(12)a^3\).
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.
\[(\Large(\text(Truncated pyramid)))\]
Definition
Consider an arbitrary pyramid \(PA_1A_2A_3...A_n\) . Let us draw a plane parallel to the base of the pyramid through a certain point lying on the side edge of the pyramid. This plane will divide the pyramid into two polyhedra, one of which is a pyramid (\(PB_1B_2...B_n\) ), and the other is called truncated pyramid(\(A_1A_2...A_nB_1B_2...B_n\) ).
The truncated pyramid has two bases - polygons \(A_1A_2...A_n\) and \(B_1B_2...B_n\) , which are similar to each other.
The height of a truncated pyramid is a perpendicular drawn from some point of the upper base to the plane of the lower base.
Important Notes
1. All side faces of a truncated pyramid are trapezoids.
2. The segment connecting the centers of the bases of a regular truncated pyramid (that is, a pyramid obtained by a section of a regular pyramid) is a height.
pyramids
Consider an arbitrary plane α, an arbitrary convex n-gon A 1 A 2 ... A n , located in this plane, and a point S that does not lie in the plane α .
Definition 1. Pyramid ( n - coal pyramid) call the figure formed by the segments connecting the point S with all points of the polygon A 1 A 2 ... A n (Fig. 1) .
Remark 1. Recall that the polygon A 1 A 2 ... A n consists of a closed broken line A 1 A 2 ... A n and the part of the plane bounded by it.
Definition 2.
Tetrahedra. Regular tetrahedra
Definition 5. An arbitrary triangular pyramid is called a tetrahedron.
Statement. For any regular triangular pyramid, opposite edges are pairwise perpendicular.
Proof. Consider a regular triangular pyramid SABC and a pair of its opposite edges, such as AC and BS . Let D denote the midpoint of the edge AC . Since segments BD and SD are medians in isosceles triangles ABC and ASC , then BD and SD are perpendicular to edge AC (Fig. 4).
where the letter D denotes the midpoint of the edge AC (Fig. 6).
By the Pythagorean theorem from the triangle BSO we find
Answer.
Formulas for volume, lateral and total surface area of a pyramid
We introduce the following notation
Then the following are true formulas for calculating the volume, area of the lateral and full surface of the pyramid:
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This video tutorial will help users to get an idea about Pyramid theme. Correct pyramid. In this lesson, we will get acquainted with the concept of a pyramid, give it a definition. Consider what a regular pyramid is and what properties it has. Then we prove the theorem on the lateral surface of a regular pyramid.
In this lesson, we will get acquainted with the concept of a pyramid, give it a definition.
Consider a polygon A 1 A 2...A n, which lies in the plane α, and a point P, which does not lie in the plane α (Fig. 1). Let's connect the dot P with peaks A 1, A 2, A 3, … A n. 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-gon A 1 A 2...A n And n triangles RA 1 A 2, RA 2 A 3 …RA n A n-1 , 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 edge.
From a point R drop the perpendicular RN on the ground plane ABCD. The perpendicular drawn is the height of the pyramid.
Rice. 2
The total surface of the pyramid consists of the lateral surface, that is, the area of all lateral faces, and the base area:
S full \u003d S side + S main
A pyramid is called correct if:
- its base is a regular polygon;
- the segment connecting the top of the pyramid with the center of the base is its height.
Explanation on the example of a regular 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 intersection point of the diagonals, is the center of the square. Means, RO is the height of the pyramid.
Rice. 3
Explanation: in the right n-gon, the center of the inscribed circle and the center of the circumscribed circle coincide. This center is called the center of the polygon. Sometimes they say that the top is projected into the center.
The height of the side face of a regular pyramid, drawn from its top, is called apothema and denoted h a.
1. all side edges of a regular pyramid are equal;
2. side faces are equal isosceles triangles.
Let us prove these properties using the example of a regular quadrangular pyramid.
Given: RABSD- regular quadrangular pyramid,
ABCD- square,
RO is the height of the pyramid.
Prove:
1. RA = PB = PC = PD
2.∆ATP = ∆BCP = ∆CDP = ∆DAP See Fig. 4.
Rice. 4
Proof.
RO is the height of the pyramid. That is, straight RO perpendicular to the plane ABC, and hence direct AO, VO, SO And DO lying in it. So the triangles ROA, ROV, ROS, ROD- rectangular.
Consider a square ABCD. It follows from the properties of a square that AO = BO = CO = DO.
Then the right triangles ROA, ROV, ROS, ROD leg RO- general and legs AO, VO, SO And DO equal, so these triangles are equal in two legs. From the equality of triangles follows the equality of segments, RA = PB = PC = PD. Point 1 is proven.
Segments AB And sun are equal because they are sides of the same square, RA = RV = PC. So the triangles AVR And VCR - isosceles and equal on three sides.
Similarly, we get that the triangles ABP, BCP, CDP, DAP are isosceles and equal, which was required to prove in item 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:
For the proof, we choose a regular triangular pyramid.
Given: RAVS is a regular triangular pyramid.
AB = BC = AC.
RO- height.
Prove: . See Fig. 5.
Rice. 5
Proof.
RAVS is a regular triangular pyramid. That is AB= AC = BC. Let ABOUT- the center of the triangle ABC, Then RO is the height of the pyramid. The base of the pyramid is an equilateral triangle. ABC. Notice, that .
triangles RAV, RVS, RSA- equal isosceles triangles (by property). A triangular pyramid has three side faces: RAV, RVS, RSA. So, the area of the lateral surface of the pyramid is:
S side = 3S RAB
The theorem has been proven.
The radius of a circle inscribed in 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- the height of the pyramid,
RO= 4 m.
Find: S side. See Fig. 6.
Rice. 6
Solution.
According to the proven theorem, .
Find the side of the base first AB. We know that the radius of a circle inscribed in 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 side DC. Because ABOUT- middle BD, volume).
Triangle DPC- isosceles. M- middle DC. That is, RM- the median, and hence the height in the triangle DPC. Then RM- apothem of the pyramid.
RO is the height of the pyramid. Then, straight RO perpendicular to the plane ABC, and hence the direct OM lying in it. Let's find an apothem RM from a right triangle ROM.
Now we can find the side surface of the pyramid:
Answer: 60 m2.
The radius of a circle circumscribed near the base of a regular triangular pyramid is m. The lateral surface area is 18 m 2. Find the length of the apothem.
Given: ABCP- regular triangular pyramid,
AB = BC = SA,
R= m,
S side = 18 m 2.
Find: . See Fig. 7.
Rice. 7
Solution.
In a right triangle ABC given the radius of the circumscribed circle. Let's find a side AB this triangle using the sine theorem.
Knowing the side of a regular triangle (m), we find its perimeter.
According to the theorem on the area of the lateral surface of a regular pyramid, where h a- apothem of the pyramid. Then:
Answer: 4 m.
So, we examined what a pyramid is, what a regular pyramid is, we proved the theorem on the lateral surface of a regular pyramid. In the next lesson, we will get acquainted with the truncated pyramid.
Bibliography
- Geometry. Grade 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th ed., Rev. and additional - M.: Mnemosyne, 2008. - 288 p.: ill.
- Geometry. Grade 10-11: A textbook for general educational institutions / Sharygin I. F. - M .: Bustard, 1999. - 208 p.: ill.
- Geometry. Grade 10: Textbook for general educational institutions with in-depth and profile 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 non-intersecting 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 is a regular triangular pyramid. Construct the linear angle of the dihedral angle at the base of the pyramid.
When a person hears the word "pyramid", he immediately recalls the majestic Egyptian structures. However, the ancient stone giants are only one of the representatives of the pyramid class. In this article, we consider from a geometric point of view the properties of a regular quadrangular pyramid.
What is a pyramid in general?
In geometry, it is understood as a three-dimensional figure, which can be obtained by connecting all the vertices of a flat polygon with one single point lying in a different plane than this polygon. The figure below shows 4 figures that satisfy this definition.
We see that the first figure has a triangular base, the second - a quadrangular one. The last two are represented by a five- and hexagonal base. However, the side surface of all pyramids is formed by triangles. Their number is exactly equal to the number of sides or vertices of the polygon at the base.
A special type of pyramids, which differ from other representatives of the class in perfect symmetry, are regular pyramids. For the figure to be correct, the following two prerequisites must be met:
- the base must be a regular polygon;
- the lateral surface of the figure should consist of equal isosceles triangles.
Note that the second mandatory condition can be replaced by another one: the perpendicular drawn to the base from the top of the pyramid (the point of intersection of the side triangles) must intersect this base in its geometric center.
Now let's move on to the topic of the article and consider what properties of a regular quadrangular pyramid characterize it. First, let's show in the figure what this figure looks like.
Its base is a square. The sides represent 4 identical isosceles triangles (they can also be equilateral with a certain ratio of the length of the side of the square and the height of the figure). The height lowered from the top of the pyramid will intersect the square in its center (the point of intersection of the diagonals).
This pyramid has 5 faces (a square and four triangles), 5 vertices (four of them belong to the base) and 8 edges. of the fourth order, passing through the height of the pyramid, translates it into itself by rotating 90 o .
The Egyptian pyramids at Giza are regular quadrangular.
Four basic linear parameters
Let's begin the consideration of the mathematical properties of a regular quadrangular pyramid with the formulas for height, length of the side of the base, side edge and apothem. Let's say right away that all these quantities are related to each other, so it is enough to know only two of them in order to unambiguously calculate the remaining two.
Suppose that the height h of the pyramid and the length a of the side of the square base are known, then the side edge b will be equal to:
b = √(a 2 / 2 + h 2)
Now we give the formula for the length a b of the apothem (the height of the triangle, lowered to the side of the base):
a b = √(a 2 / 4 + h 2)
Obviously, the side edge b is always greater than the apothem a b .
Both expressions can be used to determine all four linear characteristics if the other two parameters are known, for example a b and h.
Area and volume of a figure
These are two more important properties of a regular quadrangular pyramid. The base of the figure has the following area:
Every student knows this formula. The area of the lateral surface, which is formed by four identical triangles, can be determined through the apothem a b of the pyramid as follows:
If a b is unknown, then it can be determined by the formulas from the previous paragraph through the height h or the edge b.
The total surface area of the figure under consideration is the sum of the areas S o and S b:
S = S o + S b = a 2 + 2 × a × a b = a (a + 2 × a b)
The calculated area of all the faces of the pyramid is shown in the figure below as its sweep.
The description of the properties of a regular quadrangular pyramid will not be complete if you do not consider the formula for determining its volume. This value for the considered pyramid is calculated as follows:
That is, V is equal to the third part of the product of the figure's height and the area of \u200b\u200bits base.
Properties of a regular truncated quadrangular pyramid
You can get this figure from the original pyramid. To do this, it is necessary to cut off the upper part of the pyramid with a plane. The figure remaining under the cut plane will be called a truncated pyramid.
It is most convenient to study the characteristics of a truncated pyramid if its bases are parallel to each other. In this case, the bottom and top bases will be similar polygons. Since the base in a quadrangular regular pyramid is a square, the section formed during the cut will also be a square, but of a smaller size.
The lateral surface of the truncated figure is formed not by triangles, but by isosceles trapezoids.
One of the important properties of this pyramid is its volume, which is calculated by the formula:
V = 1/3 × h × (S o1 + S o2 + √(S o1 × S o2))
Here h is the distance between the bases of the figure, S o1, S o2 are the areas of the lower and upper bases.
- 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 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.
