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By definition, the base of a regular triangular pyramid is. What makes the pyramid a geometric miracle?

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 “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 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? 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 triangle. triangular pyramid, here we use the found formula.

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 (vertex of the pyramid) and all segments connecting the top of the pyramid with the points of the base ( lateral ribs ).

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.

Pyramid concept

Definition 1

Geometric figure, formed by a polygon and a point not lying in the plane containing this polygon, connected to all the vertices of the polygon is called a pyramid (Fig. 1).

The polygon from which the pyramid is made is called the base of the pyramid; the resulting triangles, when connected to a point, are the side faces of the pyramid, the sides of the triangles are the sides of the pyramid, and the point common to all triangles is the top of the pyramid.

Types of pyramids

Depending on the number of angles at the base of the pyramid, it can be called triangular, quadrangular, and so on (Fig. 2).

Figure 2.

Another type of pyramid is the regular pyramid.

Let us introduce and prove the property of a regular pyramid.

Theorem 1

All lateral faces of a regular pyramid are isosceles triangles that are equal to each other.

Proof.

Consider a regular $n-$gonal pyramid with vertex $S$ of height $h=SO$. Let us draw a circle around the base (Fig. 4).

Figure 4.

Consider the triangle $SOA$. According to the Pythagorean theorem, we get

Obviously, any side edge will be defined this way. Consequently, all side edges are equal to each other, that is, all side faces are isosceles triangles. Let us prove that they are equal to each other. Since the base is a regular polygon, the bases of all side faces are equal to each other. Consequently, all lateral faces are equal according to the III criterion of equality of triangles.

The theorem has been proven.

Let us now introduce the following definition related to the concept of a regular pyramid.

Definition 3

The apothem of a regular pyramid is the height of its side face.

Obviously, by Theorem One, all apothems are equal to each other.

Theorem 2

The lateral surface area of ​​a regular pyramid is determined as the product of the semi-perimeter of the base and the apothem.

Proof.

Let us denote the side of the base of the $n-$gonal pyramid by $a$, and the apothem by $d$. Therefore, the area of ​​the side face is equal to

Since, according to Theorem 1, everything sides are equal, then

The theorem has been proven.

Another type of pyramid is a truncated pyramid.

Definition 4

If a plane parallel to its base is drawn through an ordinary pyramid, then the figure formed between this plane and the plane of the base is called a truncated pyramid (Fig. 5).

Figure 5. Truncated pyramid

The lateral faces of the truncated pyramid are trapezoids.

Theorem 3

The lateral surface area of ​​a regular truncated pyramid is determined as the product of the sum of the semi-perimeters of the bases and the apothem.

Proof.

Let us denote the sides of the bases of the $n-$gonal pyramid by $a\ and\ b$, respectively, and the apothem by $d$. Therefore, the area of ​​the side face is equal to

Since all sides are equal, then

The theorem has been proven.

Sample task

Example 1

Find the area of ​​the lateral surface of a truncated triangular pyramid if it is obtained from a regular pyramid with base side 4 and apothem 5 by cutting off a plane passing through the midline of the side faces.

Solution.

By the theorem about midline we find that the upper base of the truncated pyramid is equal to $4\cdot \frac(1)(2)=2$, and the apothem is equal to $5\cdot \frac(1)(2)=2.5$.

Then, by Theorem 3, we get

Students encounter the concept of a pyramid long before studying geometry. The fault lies with the famous great Egyptian wonders of the world. Therefore, when starting to study this wonderful polyhedron, most students already clearly imagine it. All the above-mentioned attractions have the correct shape. What's happened regular pyramid, and what properties it has will be discussed further.

Definition

There are quite a lot of definitions of a pyramid. Since ancient times, it has been very popular.

For example, Euclid defined it as a bodily figure consisting of planes that, starting from one, converge at a certain point.

Heron provided a more precise formulation. He insisted that this was the figure that has a base and planes in in the form of triangles, converging at one point.

Relying on modern interpretation, the pyramid is represented as a spatial polyhedron consisting of a certain k-gon and k flat figures triangular in shape with one common point.

Let's look at it in more detail, what elements does it consist of:

  • The k-gon is considered the basis of the figure;
  • 3-gonal shapes protrude as the edges of the side part;
  • the upper part from which the side elements originate is called the apex;
  • all segments connecting a vertex are called edges;
  • if a straight line is lowered from the vertex to the plane of the figure at an angle of 90 degrees, then its part contained in the internal space is the height of the pyramid;
  • in any lateral element, a perpendicular, called an apothem, can be drawn to the side of our polyhedron.

The number of edges is calculated using the formula 2*k, where k is the number of sides of the k-gon. How many faces a polyhedron such as a pyramid has can be determined using the expression k+1.

Important! A pyramid of regular shape is a stereometric figure whose base plane is a k-gon with equal sides.

Basic properties

Correct pyramid has many properties, which are unique to her. Let's list them:

  1. The basis is a figure of the correct shape.
  2. The edges of the pyramid that limit the side elements have equal numerical values.
  3. The side elements are isosceles triangles.
  4. The base of the height of the figure falls at the center of the polygon, while it is simultaneously the central point of the inscribed and circumscribed.
  5. All side ribs are inclined to the plane of the base at the same angle.
  6. All side surfaces have the same angle of inclination relative to the base.

Thanks to all of the listed properties, performing element calculations is much simpler. Based on the above properties, we pay attention to two signs:

  1. In the case when the polygon fits into a circle, the side faces will have the base equal angles.
  2. When describing a circle around a polygon, all edges of the pyramid emanating from the vertex will have equal length and equal angles with the base.

The basis is a square

Regular quadrangular pyramid - a polyhedron whose base is a square.

It has four side faces, which are isosceles in appearance.

A square is depicted on a plane, but is based on all the properties of a regular quadrilateral.

For example, if it is necessary to relate the side of a square with its diagonal, then use the following formula: the diagonal is equal to the product of the side of the square and the square root of two.

It is based on a regular triangle

A regular triangular pyramid is a polyhedron whose base is a regular 3-gon.

If the base is a regular triangle and the side edges are equal to the edges of the base, then such a figure called a tetrahedron.

All faces of a tetrahedron are equilateral 3-gons. IN in this case You need to know some points and not waste time on them when calculating:

  • the angle of inclination of the ribs to any base is 60 degrees;
  • the size of all internal faces is also 60 degrees;
  • any face can act as a base;
  • , drawn inside the figure, these are equal elements.

Sections of a polyhedron

In any polyhedron there are several types of sections flat. Often in school course geometries work with two:

  • axial;
  • parallel to the basis.

An axial section is obtained by intersecting a polyhedron with a plane that passes through the vertex, side edges and axis. In this case, the axis is the height drawn from the vertex. The cutting plane is limited by the lines of intersection with all faces, resulting in a triangle.

Attention! In a regular pyramid, the axial section is an isosceles triangle.

If the cutting plane runs parallel to the base, then the result is the second option. In this case, we have a cross-sectional figure similar to the base.

For example, if there is a square at the base, then the section parallel to the base will also be a square, only of smaller dimensions.

When solving problems under this condition, the signs and properties of similarity of figures are used, based on Thales' theorem. First of all, it is necessary to determine the similarity coefficient.

If the plane is drawn parallel to the base and it cuts off top part polyhedron, then a regular truncated pyramid is obtained in the lower part. Then the bases of a truncated polyhedron are said to be similar polygons. In this case, the side faces are isosceles trapezoids. The axial section is also isosceles.

In order to determine the height of a truncated polyhedron, it is necessary to draw the height in the axial section, that is, in the trapezoid.

Surface areas

The main geometric problems that have to be solved in a school geometry course are finding the surface area and volume of a pyramid.

There are two types of surface area values:

  • area of ​​the side elements;
  • area of ​​the entire surface.

From the name itself it is clear what we are talking about. The side surface includes only the side elements. It follows from this that to find it, you simply need to add up the areas of the lateral planes, that is, the areas of isosceles 3-gons. Let's try to derive the formula for the area of ​​the side elements:

  1. The area of ​​an isosceles 3-gon is equal to Str=1/2(aL), where a is the side of the base, L is the apothem.
  2. The number of lateral planes depends on the type of k-gon at the base. For example, a regular quadrangular pyramid has four lateral planes. Therefore, it is necessary to add the areas of four figures Sside=1/2(aL)+1/2(aL)+1/2(aL)+1/2(aL)=1/2*4a*L. The expression is simplified in this way because the value is 4a = Rosn, where Rosn is the perimeter of the base. And the expression 1/2*Rosn is its semi-perimeter.
  3. So, we conclude that the area of ​​the lateral elements of a regular pyramid is equal to the product of the semi-perimeter of the base and the apothem: Sside = Rosn * L.

The area of ​​the total surface of the pyramid consists of the sum of the areas of the side planes and the base: Sp.p. = Sside + Sbas.

As for the area of ​​the base, here the formula is used according to the type of polygon.

Volume of a regular pyramid equal to the product of the area of ​​the base plane and the height divided by three: V=1/3*Sbas*H, where H is the height of the polyhedron.

What is a regular pyramid in geometry

Properties of the correct quadrangular 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 point R let's drop the perpendicular RN to the base plane ABCD. The drawn perpendicular is the height of the pyramid.

Rice. 2

Full surface The pyramid consists of a 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 with 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 equilateral triangle ABC. notice, that .

Triangles RAV, RVS, RSA- equal isosceles triangles (by property). A triangular pyramid has 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 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 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 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.

Bibliography

  1. 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.
  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 of 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.