What is the area of ​​the lateral surface of the truncated pyramid. Truncated pyramid

- This is a polyhedron, which is formed by the base of the pyramid and a section parallel to it. We can say that a truncated pyramid is a pyramid with a cut off top. This figure has many unique properties:

• The side faces of the pyramid are trapezoids;
• The lateral ribs of a regular truncated pyramid are of the same length and inclined to the base at the same angle;
• The bases are similar polygons;
• In a regular truncated pyramid, the faces are identical isosceles trapezoids, the area of ​​which is equal. They are also inclined to the base at one angle.

The formula for the area of ​​the lateral surface of a truncated pyramid is the sum of the areas of its sides:

Since the sides of the truncated pyramid are trapezoids, you will have to use the formula to calculate the parameters trapezoid area. For a regular truncated pyramid, another formula for calculating the area can be applied. Since all its sides, faces, and angles at the base are equal, it is possible to apply the perimeters of the base and the apothem, and also derive the area through the angle at the base.

If, according to the conditions in a regular truncated pyramid, the apothem (height of the side) and the lengths of the sides of the base are given, then the area can be calculated through the half-product of the sum of the perimeters of the bases and the apothem:

Let's look at an example of calculating the lateral surface area of ​​a truncated pyramid.
Given a regular pentagonal pyramid. Apothem l\u003d 5 cm, the length of the face in the large base is a\u003d 6 cm, and the face is at the smaller base b\u003d 4 cm. Calculate the area of ​​\u200b\u200bthe truncated pyramid.

First, let's find the perimeters of the bases. Since we are given a pentagonal pyramid, we understand that the bases are pentagons. This means that the bases are a figure with five identical sides. Find the perimeter of the larger base:

In the same way, we find the perimeter of the smaller base:

Now we can calculate the area of ​​a regular truncated pyramid. We substitute the data in the formula:

Thus, we calculated the area of ​​a regular truncated pyramid through the perimeters and apothem.

Another way to calculate lateral surface area correct pyramid, this is the formula through the corners at the base and the area of ​​\u200b\u200bthese very bases.

Let's look at an example calculation. Remember that this formula applies only to a regular truncated pyramid.

Let a regular quadrangular pyramid be given. The face of the lower base is a = 6 cm, and the face of the upper b = 4 cm. The dihedral angle at the base is β = 60°. Find the lateral surface area of ​​a regular truncated pyramid.

First, let's calculate the area of ​​the bases. Since the pyramid is regular, all the faces of the bases are equal to each other. Given that the base is a quadrilateral, we understand that it will be necessary to calculate square area. It is the product of width and length, but squared, these values ​​​​are the same. Find the area of ​​the larger base:


Now we use the found values ​​to calculate the lateral surface area.

Knowing a few simple formulas, we easily calculated the area of ​​the lateral trapezoid of a truncated pyramid through various values.

In this lesson, we will consider a truncated pyramid, get acquainted with the correct truncated pyramid, and study their properties.

Let us recall the concept of an n-gonal pyramid using the example of a triangular pyramid. Triangle ABC is given. Outside the plane of the triangle, a point P is taken, connected to the vertices of the triangle. The resulting polyhedral surface is called a pyramid (Fig. 1).

Rice. 1. Triangular pyramid

Let us cut the pyramid with a plane parallel to the plane of the base of the pyramid. The figure obtained between these planes is called a truncated pyramid (Fig. 2).

Rice. 2. Truncated pyramid

Essential elements:

Top base ;

Lower base ABC;

Side face ;

If PH is the height of the original pyramid, then is the height of the truncated pyramid.

The properties of a truncated pyramid follow from the method of its construction, namely from the parallelism of the planes of the bases:

All side faces of a truncated pyramid are trapezoids. Consider, for example, a face. It has the property of parallel planes (since the planes are parallel, they cut the side face of the original ABP pyramid along parallel lines), at the same time they are not parallel. Obviously, the quadrilateral is a trapezoid, like all the side faces of a truncated pyramid.

The ratio of the bases is the same for all trapezoids:

We have several pairs of similar triangles with the same similarity coefficient. For example, triangles and RAB are similar due to the parallelism of the planes and , the similarity coefficient:

At the same time, triangles and RCS are similar with similarity coefficient:

Obviously, the similarity coefficients for all three pairs of similar triangles are equal, so the ratio of the bases is the same for all trapezoids.

A regular truncated pyramid is a truncated pyramid obtained by cutting a regular pyramid with a plane parallel to the base (Fig. 3).

Rice. 3. Correct truncated pyramid

Definition.

A regular pyramid is called a pyramid, at the base of which lies a regular n-gon, and the vertex is projected into the center of this n-gon (the center of the inscribed and circumscribed circle).

In this case, a square lies at the base of the pyramid, and the vertex is projected to the point of intersection of its diagonals. The resulting regular quadrangular truncated pyramid has ABCD - the lower base, - the upper base. The height of the original pyramid - RO, truncated pyramid - (Fig. 4).

Rice. 4. Regular quadrangular truncated pyramid

Definition.

The height of a truncated pyramid is a perpendicular drawn from any point of one base to the plane of the second base.

The apothem of the original pyramid is RM (M is the middle of AB), the apothem of the truncated pyramid is (Fig. 4).

Definition.

The apothem of a truncated pyramid is the height of any side face.

It is clear that all the side edges of the truncated pyramid are equal to each other, that is, the side faces are equal isosceles trapezoids.

The area of ​​the lateral surface of a regular truncated pyramid is equal to the product of half the sum of the perimeters of the bases and the apothem.

Proof (for a regular quadrangular truncated pyramid - Fig. 4):

So, we need to prove:

The lateral surface area here will consist of the sum of the areas of the lateral faces - trapezoids. Since the trapezoids are the same, we have:

The area of ​​an isosceles trapezoid is the product of half the sum of the bases and the height, the apothem is the height of the trapezoid. We have:

Q.E.D.

For an n-gonal pyramid:

Where n is the number of side faces of the pyramid, a and b are the bases of the trapezoid, is the apothem.

Sides of the base of a regular truncated quadrangular pyramid are equal to 3 cm and 9 cm, height - 4 cm. Find the area of ​​the lateral surface.

Rice. 5. Illustration for problem 1

Solution. Let's illustrate the condition:

Given: , ,

Draw a straight line MN through the point O parallel to the two sides of the lower base, similarly draw a straight line through the point (Fig. 6). Since the squares and constructions are parallel at the bases of the truncated pyramid, we get a trapezoid equal to the side faces. Moreover, its lateral side will pass through the middle of the upper and lower edges of the side faces and will be the epitome of a truncated pyramid.

Rice. 6. Additional constructions

Consider the resulting trapezoid (Fig. 6). In this trapezoid, the upper base, lower base and height are known. It is required to find the lateral side, which is the apothem of the given truncated pyramid. Draw perpendicular to MN. Let us drop the perpendicular NQ from the point. We get that the larger base is divided into segments of three centimeters (). Consider a right triangle, the legs in it are known, this is an Egyptian triangle, by the Pythagorean theorem we determine the length of the hypotenuse: 5 cm.

Now there are all the elements for determining the area of ​​the lateral surface of the pyramid:

The pyramid is crossed by a plane parallel to the base. Using the example of a triangular pyramid, prove that the side edges and the height of the pyramid are divided by this plane into proportional parts.

Proof. Let's illustrate:

Rice. 7. Illustration for problem 2

The pyramid RABC is given. RO is the height of the pyramid. The pyramid is dissected by a plane, a truncated pyramid is obtained, moreover. Point - the point of intersection of the height of the RO with the plane of the base of the truncated pyramid. It is necessary to prove:

The key to the solution is the property of parallel planes. Two parallel planes cut through any third plane so that the lines of intersection are parallel. From here: . The parallelism of the corresponding lines implies the presence of four pairs of similar triangles:

From the similarity of triangles follows the proportionality of the corresponding sides. An important feature is that the similarity coefficients for these triangles are the same:

Q.E.D.

A regular triangular pyramid RABC with a height and side of the base is dissected by a plane passing through the midpoint of the height PH parallel to the base ABC. Find the area of ​​the lateral surface of the resulting truncated pyramid.

Solution. Let's illustrate:

Rice. 8. Illustration for problem 3

DIA is a regular triangle, H is the center of this triangle (the center of the inscribed and circumscribed circles). RM is the apothem of the given pyramid. - the apothem of the truncated pyramid. According to the property of parallel planes (two parallel planes cut any third plane so that the intersection lines are parallel), we have several pairs of similar triangles with an equal similarity coefficient. In particular, we are interested in the relation:

Let's find NM. This is the radius of a circle inscribed in the base, we know the corresponding formula:

Now, from the right-angled triangle РНМ, by the Pythagorean theorem, we find РМ - the apothem of the original pyramid:

From the initial ratio:

Now we know all the elements for finding the lateral surface area of ​​a truncated pyramid:

So, we got acquainted with the concepts of a truncated pyramid and a regular truncated pyramid, gave basic definitions, considered properties, and proved the theorem on the area of ​​the lateral surface. The next lesson will focus on problem solving.

Bibliography

1. I. M. Smirnova, V. A. Smirnov. Geometry. Grade 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th ed., Rev. and additional - M.: Mnemosyne, 2008. - 288 p.: ill.
2. Sharygin I. F. Geometry. Grade 10-11: Textbook for general educational institutions / Sharygin I. F. - M .: Bustard, 1999. - 208 p.: ill.
3. E. V. Potoskuev, L. I. Zvalich. Geometry. Grade 10: Textbook for general educational institutions with in-depth and profile study of mathematics / E. V. Potoskuev, L. I. Zvalich. - 6th ed., stereotype. - M.: Bustard, 2008. - 233 p.: ill.

1. Uztest.ru ().
2. Fmclass.ru ().
3. Webmath.exponenta.ru().

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