Correct pyramid of properties and notation. Geometric figures. Pyramid

  • apothem- the height of the side face of a regular pyramid, which is drawn from its vertex (in addition, the apothem is the length of the perpendicular, which is lowered from the middle of the regular polygon to one of its sides);
  • side faces (ASB, BSC, CSD, DSA) - triangles that meet at the vertex;
  • lateral ribs ( AS , B.S. , C.S. , D.S. ) — common aspects side edges;
  • top of the pyramid (t. S) - a point that connects the side ribs and which does not lie in the plane of the base;
  • height ( SO ) - a perpendicular segment 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 the pyramid- a section of the pyramid that passes through the top and the diagonal of the base;
  • base (ABCD) - a polygon that does not belong to the vertex of the pyramid.

Properties of the pyramid.

1. When all the side edges have the same size, then:

  • it is easy to describe a circle near the base of the pyramid, and the top of the pyramid will be projected into the center of this circle;
  • the side ribs form equal angles with the plane of the base;
  • Moreover, the opposite is also true, i.e. when the lateral ribs form with the plane of the base equal angles, 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, which means that all the side edges of the pyramid are the same size.

2. When the side faces have an angle of inclination to the plane of the base of the same value, then:

  • it is easy to describe a circle near the base of the pyramid, and 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 equal to ½ the product of the perimeter of the base and the height of the side face.

3. A sphere can be described around a pyramid if at the base of the pyramid there 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 middles of the edges of the pyramid perpendicular to them. From this theorem we conclude that both around any triangular and around any regular pyramid can describe the sphere.

4. A sphere can be inscribed into 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.

Based on the number of angles, the base of the pyramid is divided into triangular, quadrangular, and so on.

There will be a pyramid triangular, quadrangular, and so on, when the base of the pyramid is a triangle, a quadrangle, and so on. A triangular pyramid is a tetrahedron - a tetrahedron. Quadrangular - pentagonal and so on.

A triangular pyramid is a pyramid that has a triangle at its base. The height of this pyramid is the perpendicular that is lowered from the top of the pyramid to its base.

Finding the height of a pyramid

How to find the height of a pyramid? Very simple! To find the height of any triangular pyramid you can use the volume formula: V = (1/3)Sh, where S is the area of ​​the base, V is the volume of the pyramid, h is its height. From this formula, derive the height formula: to find the height of a triangular pyramid, you need to multiply the volume of the pyramid by 3, and then divide the resulting value by the area of ​​the base, it will be: h = (3V)/S. Since the base of a triangular pyramid is a triangle, you can use the formula to calculate the area of ​​a triangle. If we know: the area of ​​the triangle S and its side z, then according to the area formula S=(1/2)γh: h = (2S)/γ, where h is the height of the pyramid, γ is the edge of the triangle; the angle between the sides of the triangle and the two sides themselves, then using the following formula: S = (1/2)γφsinQ, where γ, φ are the sides of the triangle, we find the area of ​​the triangle. The value of the sine of angle Q needs to be looked at in the table of sines, which is available on the Internet. Next, we substitute the area value into the height formula: h = (2S)/γ. If the task requires calculating the height of a triangular pyramid, then the volume of the pyramid is already known.

Regular triangular pyramid

Find the height of a regular triangular pyramid, that is, a pyramid in which all faces are equilateral triangles, knowing the size of the edge γ. In this case, the edges of the pyramid are the sides of equilateral triangles. The height of a regular triangular pyramid will be: h = γ√(2/3), where γ is an edge equilateral triangle, h is the height of the pyramid. If the area of ​​the base (S) is unknown, and only the length of the edge (γ) and the volume (V) of the polyhedron are given, then the necessary variable in the formula from the previous step must be replaced by its equivalent, which is expressed in terms of the length of the edge. The area of ​​a triangle (regular) is equal to 1/4 of the product of the side length of this triangle squared by the square root of 3. We substitute this formula instead of the area of ​​the base in the previous formula, and we obtain the following formula: h = 3V4/(γ 2 √3) = 12V/(γ 2 √3). The volume of a tetrahedron can be expressed through the length of its edge, then from the formula for calculating the height of a figure, you can remove all variables and leave only the side of the triangular face of the figure. The volume of such a pyramid can be calculated by dividing by 12 from the product the cubed length of its face by the square root of 2.

Substituting this expression into the previous formula, we obtain the following formula for calculation: h = 12(γ 3 √2/12)/(γ 2 √3) = (γ 3 √2)/(γ 2 √3) = γ√(2 /3) = (1/3)γ√6. Also correct triangular prism can be inscribed in a sphere, and knowing only the radius of the sphere (R) one can find the height of the tetrahedron itself. The length of the edge of the tetrahedron is: γ = 4R/√6. We replace the variable γ with this expression in the previous formula and get the formula: h = (1/3)√6(4R)/√6 = (4R)/3. The same formula can be obtained by knowing the radius (R) of a circle inscribed in a tetrahedron. In this case, the length of the edge of the triangle will be equal to 12 ratios between square root of 6 and radius. We substitute this expression into the previous formula and we have: h = (1/3)γ√6 = (1/3)√6(12R)/√6 = 4R.

How to find the height of a regular quadrangular pyramid

To answer the question of how to find the length of the height of a pyramid, you need to know what a regular pyramid is. A quadrangular pyramid is a pyramid that has a quadrangle at its base. If in the conditions of the problem we have: volume (V) and area of ​​the base (S) of the pyramid, then the formula for calculating the height of the polyhedron (h) will be as follows - divide the volume multiplied by 3 by the area S: h = (3V)/S. Given a square base of a pyramid with a given volume (V) and side length γ, replace the area (S) in the previous formula with the square of the side length: S = γ 2 ; H = 3V/γ 2 . The height of a regular pyramid h = SO passes exactly through the center of the circle that is circumscribed near the base. Since the base of this pyramid is a square, point O is the intersection point of diagonals AD and BC. We have: OC = (1/2)BC = (1/2)AB√6. Next, we are in right triangle We find SOC (using the Pythagorean theorem): SO = √(SC 2 -OC 2). Now you know how to find the height of a regular 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.

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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.

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

  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 equal angles with the base.
  2. 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 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 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 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 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 sides opposite it, 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. are called side faces pyramids, segments \(PA_1, PA_2\), etc. – lateral ribs, polygon \(A_1A_2A_3A_4A_5\) – basis, point \(P\) – top.

Height pyramids are a perpendicular descended 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:

\((a)\) the lateral edges of the pyramid are equal;

\((b)\) the height of the pyramid passes through the center of the circle circumscribed near the base;

\((c)\) the side ribs are inclined to the plane of the base at the same angle.

\((d)\) the side faces are inclined to the plane of the base at the same angle.

Regular tetrahedron is a triangular pyramid, all of whose faces are equal equilateral triangles.

Theorem

Conditions \((a), (b), (c), (d)\) are equivalent.

Proof

Let's find the height of the pyramid \(PH\) . Let \(\alpha\) be the plane of the base of the pyramid.


1) Let us prove that from \((a)\) it follows \((b)\) . Let \(PA_1=PA_2=PA_3=...=PA_n\) .

Because \(PH\perp \alpha\), then \(PH\) is perpendicular to any line lying in this plane, which means the triangles are right-angled. This means that these triangles are equal in common leg \(PH\) and hypotenuse \(PA_1=PA_2=PA_3=...=PA_n\) . This means \(A_1H=A_2H=...=A_nH\) . 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 the 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 on two legs. This means that 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 both 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 TTP (\(PH\) is a perpendicular to the plane, \(HK_1, HK_2\), etc. are projections perpendicular to the sides) inclined \(PK_1, PK_2\), 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 rectangular on two sides), then the angles \(\angle PK_1H, \angle PK_2H, ...\) are equal.

5) Let us prove that \((d)\) implies \((b)\) .

Similar to the fourth point, the triangles \(PK_1H, PK_2H, ...\) are equal (as rectangular along the leg and acute angle), which means the segments \(HK_1=HK_2=...=HK_n\) are equal. This means, by definition, \(H\) is the center of a circle inscribed in the base. But because For regular polygons, the centers of the inscribed and circumscribed circles coincide, then \(H\) is the center of the circumscribed circle. Chtd.

Consequence

The lateral faces of a regular pyramid are equal isosceles triangles.

Definition

The height of the lateral face of a regular pyramid drawn from its vertex is called apothem.
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 at the point of intersection 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 at the point of intersection of the diagonals of the base (the base is a square).

3. The height of a regular hexagonal pyramid falls at 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 side edges is perpendicular to the plane of the base.


Important Notes

1. In a rectangular pyramid, the edge perpendicular to the base is the height of the pyramid. That is, \(SR\) is the height.

2. Because \(SR\) is perpendicular to any line from the base, then \(\triangle SRM, \triangle SRP\)– right triangles.

3. Triangles \(\triangle SRN, \triangle SRK\)- also rectangular.
That is, any triangle formed by this edge and the diagonal emerging from the vertex of this edge lying at the base will be rectangular.

\[(\Large(\text(Volume and surface area of ​​the pyramid)))\]

Theorem

The volume of the 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 \(V_(\text(right triangle.pir.))=\dfrac(\sqrt3)(12)a^2h\),

2. The volume of a regular quadrangular pyramid is \(V_(\text(right.four.pir.))=\dfrac13a^2h\).

3. The volume of a regular hexagonal pyramid is \(V_(\text(right.six.pir.))=\dfrac(\sqrt3)(2)a^2h\).

4. The volume of a regular tetrahedron is \(V_(\text(right tetr.))=\dfrac(\sqrt3)(12)a^3\).

Theorem

The area of ​​the lateral surface of a regular pyramid is equal to the half product of the perimeter of the base and the apothem.

\[(\Large(\text(Frustum)))\]

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 split 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 lateral 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 cross-section of a regular pyramid) is the height.

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 between 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 have one thing in common general 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

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