The volume of the truncated pyramid. Geometric figures. Truncated pyramid. For a correct truncated pyramid, the formula is correct
A pyramid is called a polyhedron, in which the base is represented by an arbitrary polygon, and the remaining faces are triangles with a common vertex, which corresponds to the vertex of the pyramid.
If a section parallel to the base is drawn in the pyramid, then it will divide the figure into two parts. The space between the lower base and the section bounded by the edges is called truncated pyramid.
The formula for the volume of a truncated pyramid is one third of the product of the height by the sum of the areas of the upper and lower bases with their average proportional:
Let's consider an example of calculating the volume of a truncated pyramid.
Problem: Given a triangular truncated pyramid. Its height is h = 10 cm, the sides of one of the bases are a = 27 cm, b = 29 cm, c = 52 cm. The perimeter of the second base is P2 = 72 cm. Find the volume of the pyramid. 
To calculate the volume, we need the area of the bases. Knowing the lengths of the sides of one triangle, we can calculate>. To do this, you need to find a semi-perimeter: 
Now let's find S2: 
Knowing that the pyramid is truncated, we conclude that the triangles lying in the bases are similar. The coefficient of similarity of these triangles can be found from the ratio of the perimeters. The ratio of the areas of the triangles will be equal to the square of this coefficient: 
Now that we've found base area truncated pyramid, we can easily calculate its volume:
Thus, having calculated the coefficient of similarity and calculating the area of the bases, we found the volume of a given truncated pyramid.
The ability to calculate the volume of spatial figures is important when solving a series practical tasks on geometry. One of the most common shapes is the pyramid. In this article, we will consider both full and truncated pyramids.
The pyramid as a three-dimensional figure
Everyone knows about Egyptian pyramids, therefore, has a good idea of which figure will be discussed. Nevertheless, Egyptian stone structures are only a special case of a huge class of pyramids.
The considered geometric object in the general case is a polygonal base, each vertex of which is connected to some point in space that does not belong to the plane of the base. This definition leads to a figure consisting of one n-gon and n triangles.
Any pyramid consists of n + 1 faces, 2 * n edges, and n + 1 vertices. Since the figure under consideration is a perfect polyhedron, the numbers of marked elements obey Euler's equality:
2 * n = (n + 1) + (n + 1) - 2.
The polygon at the base gives the name to the pyramid, for example, triangular, pentagonal, and so on. A set of pyramids with different bases is shown in the photo below.
The point at which the n triangles of the figure are connected is called the top of the pyramid. If a perpendicular is lowered from it to the base and it intersects it in the geometric center, then such a figure will be called a straight line. If this condition is not met, then an inclined pyramid takes place.
A straight figure, the base of which is formed by an equilateral (conformal) n-gon, is called regular.
The formula for the volume of a pyramid
To calculate the volume of the pyramid, we will use the integral calculus. To do this, we divide the figure with cutting planes parallel to the base into an infinite number of thin layers. The figure below shows a quadrangular pyramid with height h and side length L, in which a thin section layer is marked with a quadrangle.
The area of each such layer can be calculated using the formula:
A (z) = A 0 * (h-z) 2 / h 2.
Here A 0 is the base area, z is the value of the vertical coordinate. It can be seen that if z = 0, then the formula gives the value A 0.
To get the formula for the volume of the pyramid, you should calculate the integral over the entire height of the figure, that is:
V = ∫ h 0 (A (z) * dz).
Substituting the dependence A (z) and calculating the antiderivative, we come to the expression:
V = -A 0 * (h-z) 3 / (3 * h 2) | h 0 = 1/3 * A 0 * h.
We got the formula for the volume of the pyramid. To find the value of V, it is enough to multiply the height of the figure by the area of the base, and then divide the result by three.
Note that the resulting expression is valid for calculating the volume of a pyramid of an arbitrary type. That is, it can be inclined, and its base can be an arbitrary n-gon.
and its volume
The general formula for volume obtained in the paragraph above can be clarified in the case of a pyramid with a regular base. The area of such a base is calculated using the following formula:
A 0 = n / 4 * L 2 * ctg (pi / n).
Here L is the side length of a regular polygon with n vertices. The pi symbol is pi.
Substituting the expression for A 0 into the general formula, we get the volume of the regular pyramid:
V n = 1/3 * n / 4 * L 2 * h * ctg (pi / n) = n / 12 * L 2 * h * ctg (pi / n).
For example, for a triangular pyramid, this formula leads to the following expression:
V 3 = 3/12 * L 2 * h * ctg (60 o) = √3 / 12 * L 2 * h.
For the correct quadrangular pyramid the volume formula takes the form:
V 4 = 4/12 * L 2 * h * ctg (45 o) = 1/3 * L 2 * h.
Determining the volumes of regular pyramids requires knowing the side of their base and the height of the figure.
Truncated pyramid
Suppose that we took an arbitrary pyramid and cut off from it a part of the side surface containing the vertex. The remaining shape is called a truncated pyramid. It already consists of two n-gonal bases and n trapezoids that connect them. If the cutting plane was parallel to the base of the figure, then a truncated pyramid with parallel similar bases is formed. That is, the lengths of the sides of one of them can be obtained by multiplying the lengths of the other by some coefficient k.
The figure above demonstrates a truncated regular one.It can be seen that its upper base, like the lower one, is formed by a regular hexagon.
The formula that can be derived using a similar integral calculus is:
V = 1/3 * h * (A 0 + A 1 + √ (A 0 * A 1)).
Where A 0 and A 1 are the areas of the lower (large) and upper (small) bases, respectively. The variable h denotes the height of the truncated pyramid.
The volume of the Cheops pyramid
It is curious to solve the problem of determining the volume that the largest Egyptian pyramid contains inside itself.
In 1984, British Egyptologists Mark Lehner and Jon Goodman established the exact dimensions of the Cheops pyramid. Its original height was 146.50 meters (currently about 137 meters). The average length of each of the four sides of the structure was 230.363 meters. The base of the pyramid is square with high precision.
We will use the above figures to determine the volume of this stone giant. Since the pyramid is regular quadrangular, then the formula is valid for it:
We substitute the numbers, we get:
V 4 = 1/3 * (230.363) 2 * 146.5 ≈ 2591444 m 3.
The volume of the Cheops pyramid is almost 2.6 million m 3. For comparison, we note that the Olympic pool has a volume of 2.5 thousand m 3. That is, to fill the entire Cheops pyramid, more than 1000 such pools will be needed!
and a cutting plane that is parallel to its base.
Or in other words: truncated pyramid- this is such a polyhedron, which is formed by a pyramid and its section parallel to the base.
A section parallel to the base of the pyramid divides the pyramid into 2 parts. The part of the pyramid between its base and section is truncated pyramid.
This section for the truncated pyramid turns out to be one of the bases of this pyramid.
The distance between the bases of the truncated pyramid is truncated pyramid height.
The truncated pyramid will correct when the pyramid from which it was obtained was also correct.
The height of the trapezoid of the side face of a regular truncated pyramid is apothem the correct truncated pyramid.
Truncated pyramid properties.
1. Each side face of a regular truncated pyramid is isosceles trapezoids of the same size.
2. The bases of the truncated pyramid are similar polygons.
3. The side edges of a regular truncated pyramid are of equal size and one is inclined in relation to the base of the pyramid.
4. The side faces of the truncated pyramid are trapeziums.
5. The dihedral angles at the lateral edges of a regular truncated pyramid are of equal magnitude.
6. The ratio of the areas of the bases: S 2 / S 1 = k 2.
Truncated pyramid formulas.
For an arbitrary pyramid:
The volume of the truncated pyramid is equal to 1/3 of the product of the height h (OS) for the sum of the areas of the upper base S 1 (abcde), the lower base of the truncated pyramid S 2 (ABCDE) and the average proportional between them.
Pyramid volume:
where S 1, S 2- the area of the bases,
h- the height of the truncated pyramid.
Lateral surface area 
equal to the sum of the areas of the side faces of the truncated pyramid.
For a correct truncated pyramid:
Correct truncated pyramid- a polyhedron, which is formed by a regular pyramid and its section, which is parallel to the base.
The lateral surface area of a regular truncated pyramid is ½ of the product of the sum of the perimeters of its bases and the apothem.
where S 1, S 2- the area of the bases,
φ - dihedral angle at the base of the pyramid.
CH is the height of the truncated pyramid, P 1 and P 2- perimeters of the bases, S 1 and S 2- the areas of the bases, S side- lateral surface area, S full- total surface area:
Section of the pyramid with a plane parallel to the base.
Section of the pyramid by a plane parallel to its base (perpendicular to the height) divides the height and side edges of the pyramid into proportional segments.
The section of the pyramid by a plane that is parallel to its base (perpendicular to the height) is a polygon that is similar to the base of the pyramid, while the coefficient of similarity of these polygons corresponds to the ratio of their distances from the top of the pyramid.
The areas of the sections that are parallel to the base of the pyramid are related as the squares of their distances from the top of the pyramid.
Pyramid is called a polyhedron, one of whose faces is a polygon ( base ), and all other faces are triangles with a common vertex ( side faces ) (fig. 15). The pyramid is called correct , if its base is a regular polygon and the top of the pyramid is projected to the center of the base (Fig. 16). A triangular pyramid in which all edges are equal is called tetrahedron .
Side rib pyramid is the side of the side face that does not belong to the base Height pyramid is called the distance from its top to the plane of the base. All side edges of a regular pyramid are equal to each other, all side edges are equal isosceles triangles... The height of the side face of a regular pyramid drawn from the top is called apothem . Diagonal section the section of the pyramid is called a plane passing through two lateral edges that do not belong to one face.
Lateral surface area pyramid is called the sum of the areas of all side faces. Full surface area called the sum of the areas of all side faces and the base.
Theorems
1. If in a pyramid all lateral edges are equally inclined to the plane of the base, then the top of the pyramid is projected into the center of the circle circumscribed about the base.
2. If in the pyramid all side edges have equal lengths, then the top of the pyramid is projected into the center of the circle circumscribed about the base.
3. If in the pyramid all the faces are equally inclined to the plane of the base, then the top of the pyramid is projected into the center of the circle inscribed in the base.
To calculate the volume of an arbitrary pyramid, the following formula is correct:
where V- volume;
S main- base area;
H- the height of the pyramid.
For the correct pyramid, the formulas are correct:
where p- base perimeter;
h a- apothem;
H- height;
S full
S side
S main- base area;
V- the volume of the correct pyramid.
Truncated pyramid called the part of the pyramid, enclosed between the base and the secant plane parallel to the base of the pyramid (Fig. 17). Regular truncated pyramid is called the part of a regular pyramid, enclosed between the base and the secant plane parallel to the base of the pyramid.
Foundations truncated pyramids - similar polygons. Side faces - trapezoid. Height a truncated pyramid is the distance between its bases. Diagonal a truncated pyramid is called a segment connecting its vertices that do not lie on the same face. Diagonal section a section of a truncated pyramid is called a plane passing through two lateral edges that do not belong to one face.
For a truncated pyramid, the following formulas are valid:
(4)
where S 1 , S 2 - areas of the upper and lower bases;
S full- total surface area;
S side- lateral surface area;
H- height;
V- the volume of the truncated pyramid.
For a correct truncated pyramid, the formula is correct:
where p 1 , p 2 - perimeters of the bases;
h a- the apothem of the regular truncated pyramid.
Example 1. In the correct triangular pyramid the dihedral angle at the base is 60º. Find the tangent of the angle of inclination of the side edge to the plane of the base.
Solution. Let's make a drawing (fig. 18).
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The pyramid is regular, so at the base there is an equilateral triangle and all side faces are equal isosceles triangles. The dihedral angle at the base is the angle of inclination of the side face of the pyramid to the plane of the base. The linear angle is the angle a between two perpendiculars: and i.e. The top of the pyramid is projected in the center of the triangle (the center of the circumcircle and the inscribed circle in the triangle ABC). The angle of inclination of the lateral rib (for example SB) Is the angle between the edge itself and its projection onto the plane of the base. For rib SB this angle will be the angle SBD... To find the tangent, you need to know the legs SO and OB... Let the length of the segment BD is equal to 3 a... Dot O section BD is divided into parts: and From we find SO: 
From we find: 
Answer:
Example 2. Find the volume of a regular truncated quadrangular pyramid if the diagonals of its bases are cm and cm, and the height is 4 cm.
Solution. To find the volume of the truncated pyramid, we use formula (4). To find the area of the bases, you need to find the sides of the base squares, knowing their diagonals. The sides of the bases are 2 cm and 8 cm, respectively. So the areas of the bases and Having substituted all the data in the formula, we calculate the volume of the truncated pyramid:
Answer: 112 cm 3.
Example 3. Find the area of the side face of a regular triangular truncated pyramid, the sides of the bases of which are 10 cm and 4 cm, and the height of the pyramid is 2 cm.
Solution. Let's make a drawing (fig. 19).
The side face of this pyramid is an isosceles trapezoid. To calculate the area of a trapezoid, you need to know the base and height. The bases are given by condition, only the height remains unknown. We will find it from where A 1 E perpendicular from point A 1 on the plane of the lower base, A 1 D- perpendicular from A 1 on AS. A 1 E= 2 cm, since this is the height of the pyramid. To find DE let's make an additional drawing, which will depict a top view (fig. 20). Dot O- projection of the centers of the upper and lower bases. since (see fig. 20) and On the other hand OK Is the radius of the inscribed circle and 
OM- radius of the inscribed circle:
MK = DE.
By the Pythagorean theorem from
Side face area: 
Answer:
Example 4. At the base of the pyramid lies an isosceles trapezoid, the bases of which a and b (a> b). Each side face forms an angle with the base plane of the pyramid equal to j... Find the total surface area of the pyramid.
Solution. Let's make a drawing (fig. 21). Total surface area of the pyramid SABCD equal to the sum of the areas and area of the trapezoid ABCD.
Let us use the statement that if all the faces of the pyramid are equally inclined to the plane of the base, then the apex is projected to the center of the circle inscribed in the base. Dot O- vertex projection S at the base of the pyramid. Triangle SOD is the orthogonal projection of the triangle CSD on the plane of the base. By the theorem on the area of an orthogonal projection of a plane figure, we get:
Similarly, it means 
Thus, the task was reduced to finding the area of the trapezoid ABCD... Draw a trapezoid ABCD separately (fig. 22). Dot O- the center of the circle inscribed in the trapezoid.
Since a circle can be inscribed in a trapezoid, either From, by the Pythagorean theorem, we have
