The correct pyramid of properties and formulas. Formulas and properties of a regular quadrangular pyramid
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.
Here are collected basic information about the pyramids and related formulas and concepts. All of them are studied with a tutor in mathematics in preparation for the exam.
Consider a plane, a polygon 
lying in it and a point S not lying in it. Connect S to all vertices of the polygon. The resulting polyhedron is called a pyramid. The segments are called lateral edges. The polygon is called the base, and the point S is called 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. Alternative name for the triangular pyramid - tetrahedron. The height of a pyramid is the perpendicular drawn from its apex to the base plane.
A pyramid is called correct if a regular polygon, and the base of the height of the pyramid (the base of the perpendicular) is its center.
Tutor's comment:
Do not confuse the concept 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 of the edges are equal. This is his definition. It is easy to prove that the equality implies that the center P of the polygon with a height base, 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.
Mathematics tutor about his terminology: work with pyramids is 80% 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 name the first of them apothemic, and 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 total surface area of the pyramid.
3) , where MN is the distance of any two crossing edges, and is the area of the parallelogram formed by the midpoints of the four remaining edges.
Pyramid Height Base Property:
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 towards 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 commentary: note that all points are united by one common property: one way or another, side faces participate everywhere (apothems are their elements). Therefore, the tutor can offer a less precise, but more convenient formulation for memorization: the 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 suffices to show that all apothemical triangles are equal.
The point P coincides with the center of the circumscribed circle near the base of the pyramid, if one of the three conditions is true:
1) All side edges are equal
2) All side ribs are equally inclined towards the base
3) All side ribs are equally inclined to the height
Important notes!
1. If instead of formulas you see abracadabra, clear the cache. How to do it in your browser is written here:
2. Before you start reading the article, pay attention to our navigator for the most useful resource for
What is a pyramid?
How she looks like?
You see: at the pyramid below (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 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, quite "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;
At the same time, the point where it fell height, is called height base. Note that in the "crooked" pyramids height may even be outside the pyramid. Like this:
And there is nothing terrible in this. It looks like an obtuse triangle.
Correct pyramid.
Lots of difficult words? Let's decipher: " At the base - correct"- this is understandable. And now remember that a regular polygon has a center - a point that is the center of and , and .
Well, and 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 right pyramid.
Hexagonal: at the base - a regular hexagon, the vertex is projected into the center of the base.
quadrangular: at the base - a square, the top is projected to the intersection point of the diagonals of this square.
triangular: at the base is a regular triangle, the vertex is projected to the intersection point 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 side faces are isosceles triangles and all these triangles are equal.
Pyramid Volume
The main formula for the volume of the pyramid:
Where did it come from exactly? This is not so simple, and at first you just need to remember that the pyramid and cone have volume in the formula, but the 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. I need to find and.
This is the area of a right triangle.
Let's remember how to search for this area. We use the area formula:
We have "" - this, and "" - this too, eh.
Now let's find.
According to the Pythagorean theorem for
What does it matter? This is the radius of the circumscribed circle in, because pyramidcorrect and hence the center.
Since - the point of intersection and the median too.
(Pythagorean theorem for)
Substitute in the formula for.
Let's plug everything into the volume formula:
Attention: if you have a regular tetrahedron (i.e.), then the formula is:
Let the side of the base be equal, and the side edge equal.
There is no need to search here; because at the base is a square, and therefore.
Let's find. According to the Pythagorean theorem for
Do we know? Almost. Look:
(we saw this by reviewing).
Substitute in 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 searched for the area of a regular triangle when calculating the volume of a regular triangular pyramid, here we use the found formula.
Now let's find (this).
According to the Pythagorean theorem for
But what does it matter? It's simple because (and everyone else too) is correct.
We substitute:
\displaystyle V=\frac(\sqrt(3))(2)((a)^(2))\sqrt(((b)^(2))-((a)^(2)))
PYRAMID. BRIEFLY ABOUT THE MAIN
A pyramid is a polyhedron that consists of any flat polygon (), a point that does not lie in the plane of the base (top of the pyramid) and all segments connecting the top of the pyramid to the 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, which has a regular polygon 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 side edges are equal.
- All side faces are isosceles triangles and all these triangles are equal.
Volume of the pyramid:
Well, the topic is over. If you are reading these lines, then you are very cool.
Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!
Now the most important thing.
You've figured out the theory on this topic. And, I repeat, it's ... it's just super! You are already better than the vast majority of your peers.
The problem is that this may not be enough ...
For what?
For the successful passing of the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.
I will not convince you of anything, I will just say one thing ...
People who have received a good education earn much more than those who have not received it. This is statistics.
But this is not the main thing.
The main thing is that they are MORE HAPPY (there are such studies). Perhaps because much more opportunities open up before them and life becomes brighter? Do not know...
But think for yourself...
What does it take to be sure to be better than others on the exam and be ultimately ... happier?
FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.
On the exam, you will not be asked theory.
You will need solve problems on time.
And, if you haven’t solved them (LOTS!), you will definitely make a stupid mistake somewhere or simply won’t make it in time.
It's like in sports - you need to repeat many times to win for sure.
Find a collection anywhere you want necessarily with solutions, detailed analysis and decide, decide, decide!
You can use our tasks (not necessary) and we certainly recommend them.
In order to get a hand with the help of our tasks, you need to help extend the life of the YouClever textbook that you are currently reading.
How? There are two options:
- Unlock access to all hidden tasks in this article -
- Unlock access to all hidden tasks in all 99 articles of the tutorial - Buy a textbook - 499 rubles
Yes, we have 99 such articles in the textbook and access to all tasks and all hidden texts in them can be opened immediately.
Access to all hidden tasks is provided for the entire lifetime of the site.
In conclusion...
If you don't like our tasks, find others. Just don't stop with theory.
“Understood” and “I know how to solve” are completely different skills. You need both.
Find problems and solve!
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 etc.
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. one.
Rice. one
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. five.
Rice. five
Proof.
RAVS is a regular triangular pyramid. I.e AB= AC = BC. Let be 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.
To 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 be M- middle side DC. Because ABOUT- middle BD, volume).
Triangle DPC- isosceles. M- middle DC. I.e, 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.
To 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: 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.
- 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.