Quadrangular pyramid in problem C2. Basics of geometry: a regular pyramid is

When solving Problem C2 using the coordinate method, many students face the same problem. They can't calculate coordinates of points included in the scalar product formula. The greatest difficulties arise pyramids. And if the base points are considered more or less normal, then the tops are a real hell.

Today we will work on a regular quadrangular pyramid. There is also a triangular pyramid (aka - tetrahedron). This is a more complex design, so a separate lesson will be devoted to it.

First, let's remember the definition:

A regular pyramid is one that:

The base is a regular polygon: triangle, square, etc.;

An altitude drawn to the base passes through its center.

In particular, the base of a quadrangular pyramid is square. Just like Cheops, only a little smaller.

Below are calculations for a pyramid in which all edges are equal to 1. If this is not the case in your problem, the calculations do not change - just the numbers will be different.

Vertices of a quadrangular pyramid

So, let a regular quadrangular pyramid SABCD be given, where S is the vertex and the base ABCD is a square. All edges are equal to 1. You need to enter a coordinate system and find the coordinates of all points. We have:

We introduce a coordinate system with origin at point A:

The OX axis is directed parallel to the edge AB;
OY axis is parallel to AD. Since ABCD is a square, AB ⊥ AD;
Finally, we direct the OZ axis upward, perpendicular to the ABCD plane.

Now we calculate the coordinates. Additional construction: SH - height drawn to the base. For convenience, we will place the base of the pyramid in a separate drawing. Since points A, B, C and D lie in the OXY plane, their coordinate is z = 0. We have:

A = (0; 0; 0) - coincides with the origin;
B = (1; 0; 0) - step by 1 along the OX axis from the origin;
C = (1; 1; 0) - step by 1 along the OX axis and by 1 along the OY axis;
D = (0; 1; 0) - step only along the OY axis.
H = (0.5; 0.5; 0) - the center of the square, the middle of the segment AC.

It remains to find the coordinates of point S. Note that the x and y coordinates of points S and H are the same, since they lie on a line parallel to the OZ axis. It remains to find the z coordinate for point S.

Consider triangles ASH and ABH:

AS = AB = 1 by condition;
Angle AHS = AHB = 90°, since SH is the height and AH ⊥ HB as the diagonals of the square;
Side AH is common.

Therefore, right triangles ASH and ABH equal one leg and one hypotenuse each. This means SH = BH = 0.5 BD. But BD is the diagonal of a square with side 1. Therefore we have:

Total coordinates of point S:

In conclusion, we write down the coordinates of all the vertices of a regular rectangular pyramid:

What to do when the ribs are different

What if the side edges of the pyramid are not equal to the edges of the base? In this case, consider the triangle AHS:

Triangle AHS - rectangular, and the hypotenuse AS is also a side edge of the original pyramid SABCD. Leg AH is easily calculated: AH = 0.5 AC. We will find the remaining leg SH according to the Pythagorean theorem. This will be the z coordinate for point S.

Task. Given a regular quadrangular pyramid SABCD, at the base of which lies a square with side 1. Side edge BS = 3. Find the coordinates of point S.

We already know the x and y coordinates of this point: x = y = 0.5. This follows from two facts:

The projection of point S onto the OXY plane is point H;
At the same time, point H is the center of a square ABCD, all sides of which are equal to 1.

It remains to find the coordinate of point S. Consider triangle AHS. It is rectangular, with the hypotenuse AS = BS = 3, the leg AH being half the diagonal. For further calculations we need its length:

Pythagorean theorem for triangle AHS: AH 2 + SH 2 = AS 2. We have:

So, the coordinates of point S.

Quadrangular pyramid is a polyhedron whose base is a square, and all its side faces are identical isosceles triangles.

This polyhedron has many different properties:

Its lateral edges and adjacent dihedral angles are equal to each other;
The areas of the side faces are the same;
At the base of a regular quadrangular pyramid lies a square;
The height dropped from the top of the pyramid intersects the point where the diagonals of the base intersect.

All these properties make it easy to find. However, quite often, in addition to this, it is necessary to calculate the volume of the polyhedron. To do this, use the formula for the volume of a quadrangular pyramid:

That is, the volume of the pyramid is equal to one third of the product of the height of the pyramid and the area of the base. Since it is equal to the product of its equal sides, we immediately enter the formula for the area of a square into the expression for volume.
Let's consider an example of calculating the volume of a quadrangular pyramid.

Let a quadrangular pyramid be given, the base of which is a square with side a = 6 cm. The side face of the pyramid is b = 8 cm. Find the volume of the pyramid.

To find the volume of a given polyhedron, we need the length of its height. Therefore, we will find it by applying the Pythagorean theorem. First, let's calculate the length of the diagonal. In the blue triangle it will be the hypotenuse. It is also worth remembering that the diagonals of a square are equal to each other and are divided in half at the point of intersection:

Now from the red triangle we find the height h we need. It will be equal to:

Let's substitute the necessary values and find the height of the pyramid:

Now, knowing the height, we can substitute all the values into the formula for the volume of the pyramid and calculate the required value:

In this way, knowing a few simple formulas, we were able to calculate the volume of a regular quadrangular pyramid. Remember that this value is measured in cubic units.

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 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). 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, volume).

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 a 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 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

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. Grades 10-11: Textbook for general education institutions / Sharygin I. F. - M.: Bustard, 1999. - 208 pp.: 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.

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.

In contact with

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 the form of triangles, converging at one point.

Based on the modern interpretation, the pyramid is represented as a spatial polyhedron, consisting of a certain k-gon and k flat triangular figures, having 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:

The basis is a figure of the correct shape.
The edges of the pyramid that limit the side elements have equal numerical values.
The side elements are isosceles triangles.
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.
All side ribs are inclined to the plane of the base at the same angle.
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:

In the case when the polygon fits into a circle, the side faces will have equal angles with the base.
When describing a circle around a polygon, all edges of the pyramid emanating from the vertex will have equal lengths 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 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 a school geometry course they 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, they use signs and properties of similarity of figures, 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 the upper part of the 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:

The area of an isosceles 3-gon is Str=1/2(aL), where a is the side of the base, L is the apothem.
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.
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 a regular quadrangular pyramid

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 second costal. Unfortunately, you will not find this terminology in any of the textbooks, and the teacher has to introduce it unilaterally.

Formula for the volume of a pyramid:
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 of 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