How to find the base of a height in a pyramid. Basic properties of a regular pyramid

Video tutorial 2: Pyramid problem. Pyramid volume

Video tutorial 3: Pyramid problem. Correct pyramid

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

Pyramid, its properties

Pyramid- This volumetric body, which 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– S. That is, this means that everything 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 side surface of the pyramid consists of triangles, to find the total area of the side surface it is necessary 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.
At the same time sufficient condition In addition to the fact that a circle can be described around the base of the pyramid, and we can also assume that all the edges are of different lengths, we can consider 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 the base of the pyramid lies regular polygon(With equal sides), then such a pyramid will be called regular.

Regular triangular pyramid

A three-dimensional figure that often appears in geometric problems is the pyramid. The simplest of all the figures in this class is triangular. In this article we will analyze in detail basic formulas and properties of the correct

Geometric ideas about the figure

Before moving on to the properties regular pyramid triangular, let’s take a closer look at what kind of figure we are talking about.

Let's assume that there is arbitrary triangle V three-dimensional space. Let us select any point in this space that does not lie in the plane of the triangle and connect it with the three vertices of the triangle. We got a triangular pyramid.

It consists of 4 sides, all of which are triangles. The points where three faces meet are called vertices. The figure also has four of them. The lines of intersection of two faces are edges. The pyramid in question has 6 edges. The figure below shows an example of this figure.

Since the figure is formed by four sides, it is also called a tetrahedron.

Correct pyramid

Above we considered an arbitrary figure with a triangular base. Now suppose we have perpendicular segment from the top of the pyramid to its base. This segment is called height. Obviously, you can draw 4 different heights for the figure. If the height intersects at geometric center triangular base, then such a pyramid is called straight.

A straight pyramid, the base of which is an equilateral triangle, is called regular. For her, all three triangles forming the lateral surface of the figure are isosceles and equal to each other. A special case of a regular pyramid is the situation when all four sides are equilateral identical triangles.

Let's consider the properties of a regular triangular pyramid and give the corresponding formulas for calculating its parameters.

Base side, height, lateral edge and apothem

Any two of the listed parameters uniquely determine the remaining two characteristics. Let us present formulas that relate these quantities.

Let us assume that the side of the base of a regular triangular pyramid is a. The length of its lateral edge is b. What will be the height of a regular triangular pyramid and its apothem?

For height h we get the expression:

This formula follows from the Pythagorean theorem for which the side edge, the height and 2/3 of the height of the base are.

The apothem of a pyramid is the height for any side triangle. The length of the apothem a b is equal to:

a b = √(b 2 - a 2 /4)

From these formulas it is clear that whatever the side of the base of a triangular regular pyramid and the length of its side edge, the apothem will always be more height pyramids.

The two formulas presented contain all four linear characteristics of the figure in question. Therefore, given the known two of them, you can find the rest by solving the system of written equalities.

Figure volume

For absolutely any pyramid (including an inclined one), the value of the volume of space limited by it can be determined by knowing the height of the figure and the area of its base. Corresponding formula has the form:

Applying this expression to the figure under consideration, we obtain the following formula:

Where the height of a regular triangular pyramid is h and its base side is a.

It is not difficult to obtain a formula for the volume of a tetrahedron in which all sides are equal to each other and represent equilateral triangles. In this case, the volume of the figure is determined by the formula:

That is, it is determined uniquely by the length of side a.

Surface area

Let us continue to consider the properties of a regular triangular pyramid. Total area of all the faces of a figure is called its surface area. The latter can be conveniently studied by considering the corresponding development. The figure below shows what the development of a regular triangular pyramid looks like.

Let's assume that we know the height h and the side of the base a of the figure. Then the area of its base will be equal to:

Every schoolchild can obtain this expression if he remembers how to find the area of a triangle and also takes into account that the height equilateral triangle is also a bisector and a median.

The area of the lateral surface formed by three identical isosceles triangles, is:

S b = 3/2*√(a 2 /12+h 2)*a

This equality follows from the expression of the apothem of the pyramid in terms of the height and length of the base.

The total surface area of the figure is:

S = S o + S b = √3/4*a 2 + 3/2*√(a 2 /12+h 2)*a

Note that for a tetrahedron in which all four sides are identical equilateral triangles, the area S will be equal to:

Properties of a regular truncated triangular pyramid

If the considered triangular pyramid has a plane, parallel to the base, cut off the top, then the remaining lower part will be called a truncated pyramid.

In the case of a triangular base, the result of the described sectioning method is a new triangle, which is also equilateral, but has a shorter side length than the side of the base. A truncated triangular pyramid is shown below.

We see that this figure is already limited to two triangular bases and three isosceles trapezoids.

Let us assume that the height of the resulting figure is equal to h, the lengths of the sides of the lower and upper bases are a 1 and a 2, respectively, and the apothem (height of the trapezoid) is equal to a b. Then the surface area of the truncated pyramid can be calculated using the formula:

S = 3/2*(a 1 +a 2)*a b + √3/4*(a 1 2 + a 2 2)

Here the first term is the area of the lateral surface, the second term is the area of the triangular bases.

The volume of the figure is calculated as follows:

V = √3/12*h*(a 1 2 + a 2 2 + a 1 *a 2)

For unambiguous definition characteristics of a truncated pyramid, you need to know its three parameters, which is demonstrated by the given formulas.

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 edge size 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 the edge of the 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 the figure, you can remove all variables and leave only the side triangular face figures. 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: the volume (V) and the 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/γ2. 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.

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.

Based on the 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 enclosed in internal space— 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! Pyramid correct form called 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 everyone listed properties, performing element calculations is much easier. 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 the base equal angles.
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 right 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 the base is a square, 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 axial section, that is, in a trapezoid.

Surface areas

Basic 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. Lateral surface includes only 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 equal to 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, the correct quadrangular pyramid has four lateral planes. Therefore, it is necessary to add area 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.

Square full surface 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