Triangular pyramid formula. Basic properties of a regular pyramid

Video tutorial 2: Pyramid problem. Volume of the pyramid

Video tutorial 3: Pyramid problem. Correct pyramid

Lecture: Pyramid, its base, lateral ribs, height, lateral surface; triangular pyramid; regular pyramid

Pyramid, its properties

Pyramid is a three-dimensional body that has a polygon at its base, and all its faces consist of triangles.

A special case of a pyramid is a cone with a circle at its base.


Let's look at the main elements of the pyramid:


Apothem- this is a segment that connects the top of the pyramid with the middle of the lower edge of the side face. In other words, this is the height of the edge of the pyramid.


In the figure you can see triangles ADS, ABS, BCS, CDS. If you look closely at the names, you can see that each triangle has one common letter in its name - S. That is, this means that all the side faces (triangles) converge at one point, which is called the top of the pyramid.


The segment OS that connects the vertex with the point of intersection of the diagonals of the base (in the case of triangles - at the point of intersection of the heights) is called pyramid height.


A diagonal section is a plane that passes through the top of the pyramid, as well as one of the diagonals of the base.


Since the lateral surface of the pyramid consists of triangles, then to find total area side surface, you need to find the area of ​​each face and add them up. The number and shape of faces depends on the shape and size of the sides of the polygon that lies at the base.


The only plane in a pyramid that does not belong to its vertex is called basis pyramids.

In the figure we see that the base is a parallelogram, however, it can be any arbitrary polygon.

Properties:


Consider the first case of a pyramid, in which it has edges of the same length:

  • A circle can be drawn around the base of such a pyramid. If you project the top of such a pyramid, then its projection will be located in the center of the circle.
  • The angles at the base of the pyramid are the same on each face.
  • In this case, a sufficient condition for the fact that a circle can be described around the base of the pyramid, and also that all the edges are of different lengths, can be considered the same angles between the base and each edge of the faces.

If you come across a pyramid in which the angles between the side faces and the base are equal, then the following properties are true:

  • You will be able to describe a circle around the base of the pyramid, the apex of which is projected exactly at the center.
  • If you draw each side edge of the height to the base, then they will be of equal length.
  • To find the lateral surface area of ​​such a pyramid, it is enough to find the perimeter of the base and multiply it by half the length of the height.
  • S bp = 0.5P oc H.
  • Types of pyramid.
  • Depending on which polygon lies at the base of the pyramid, they can be triangular, quadrangular, etc. If at the base of the pyramid lies a regular polygon (with equal sides), then such a pyramid will be called regular.

Regular triangular pyramid

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 the 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 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 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. all lateral edges of a regular pyramid are equal;

2. The side faces are equal 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 an equilateral triangle ABC. notice, 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 the 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 right triangle ROM.

Now we can find the lateral surface of the pyramid:

Answer: 60 m2.

The radius of the circle circumscribed around the base of a regular triangular pyramid is equal to 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 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 regular triangle(m), let's 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.

Bibliography

  1. Geometry. Grades 10-11: textbook for students educational institutions(basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th ed., rev. and additional - M.: Mnemosyne, 2008. - 288 p.: ill.
  2. Geometry. 10-11 grade: Textbook for general education educational institutions/ Sharygin I.F. - M.: Bustard, 1999. - 208 p.: ill.
  3. 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.
  1. Internet portal "Yaklass" ()
  2. Internet portal "Festival pedagogical ideas"First of September" ()
  3. Internet portal “Slideshare.net” ()

Homework

  1. Can a regular polygon be the base of an irregular pyramid?
  2. Prove that disjoint edges of a regular pyramid are perpendicular.
  3. 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.
  4. RAVS- regular triangular pyramid. Construct the linear angle of the dihedral angle at the base of the pyramid.

We continue to consider the tasks included in the Unified State Examination in mathematics. We have already studied problems where the condition is given and it is required to find the distance between two given points or an angle.

A pyramid is a polyhedron, the base of which is a polygon, the remaining faces are triangles, and they have a common vertex.

A regular pyramid is a pyramid at the base of which lies a regular polygon, and its vertex is projected into the center of the base.

A regular quadrangular pyramid - the base is a square. The top of the pyramid is projected at the point of intersection of the diagonals of the base (square).


ML - apothem
∠MLO - dihedral angle at the base of the pyramid
∠MCO - angle between the lateral edge and the plane of the base of the pyramid

In this article we will look at problems to solve a regular pyramid. You need to find some element, lateral surface area, volume, height. Of course, you need to know the Pythagorean theorem, the formula for the area of ​​the lateral surface of a pyramid, and the formula for finding the volume of a pyramid.

In the article "" presents the formulas that are necessary to solve problems in stereometry. So, the tasks:

SABCD dot O- center of the base,S vertex, SO = 51, A.C.= 136. Find the side edgeS.C..

IN in this case the base is a square. This means that the diagonals AC and BD are equal, they intersect and are bisected by the intersection point. Note that in a regular pyramid the height dropped from its top passes through the center of the base of the pyramid. So SO is the height and the triangleSOCrectangular. Then according to the Pythagorean theorem:

How to extract the root from large number.

Answer: 85

Decide for yourself:

In a regular quadrangular pyramid SABCD dot O- center of the base, S vertex, SO = 4, A.C.= 6. Find the side edge S.C..

In a regular quadrangular pyramid SABCD dot O- center of the base, S vertex, S.C. = 5, A.C.= 6. Find the length of the segment SO.

In a regular quadrangular pyramid SABCD dot O- center of the base, S vertex, SO = 4, S.C.= 5. Find the length of the segment A.C..

SABC R- middle of the rib B.C., S- top. It is known that AB= 7, a S.R.= 16. Find the lateral surface area.

The area of ​​the lateral surface of a regular triangular pyramid is equal to half the product of the perimeter of the base and the apothem (apothem is the height of the lateral face of a regular pyramid drawn from its vertex):

Or we can say this: the area of ​​the lateral surface of the pyramid is equal to the sum three squares side edges. The lateral faces in a regular triangular pyramid are triangles of equal area. In this case:

Answer: 168

Decide for yourself:

In a regular triangular pyramid SABC R- middle of the rib B.C., S- top. It is known that AB= 1, a S.R.= 2. Find the lateral surface area.

In a regular triangular pyramid SABC R- middle of the rib B.C., S- top. It is known that AB= 1, and the area of ​​the lateral surface is 3. Find the length of the segment S.R..

In a regular triangular pyramid SABC L- middle of the rib B.C., S- top. It is known that SL= 2, and the area of ​​the lateral surface is 3. Find the length of the segment AB.

In a regular triangular pyramid SABC M. Area of ​​a triangle ABC is 25, the volume of the pyramid is 100. Find the length of the segment MS.

The base of the pyramid is an equilateral triangle. That's why Mis the center of the base, andMS- height of a regular pyramidSABC. Volume of the pyramid SABC equals: view solution

In a regular triangular pyramid SABC the medians of the base intersect at the point M. Area of ​​a triangle ABC equals 3, MS= 1. Find the volume of the pyramid.

In a regular triangular pyramid SABC the medians of the base intersect at the point M. The volume of the pyramid is 1, MS= 1. Find the area of ​​the triangle ABC.

Let's finish here. As you can see, problems are solved in one or two steps. In the future, we will consider other problems from this part, where bodies of revolution are given, don’t miss it!

I wish you success!

Sincerely, Alexander Krutitskikh.

P.S: I would be grateful if you tell me about the site on social networks.

First level

Pyramid. Visual Guide (2019)

What is a pyramid?

How does she look?

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.

A lot of complex 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 of ​​a 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? 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.
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