The formula for the volume of a truncated pyramid. Volume formulas for a full and truncated pyramid
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The ability to calculate the volume of spatial figures is important when solving a series practical tasks by geometry. One of the most common shapes is the pyramid. In this article, we will consider the pyramids, both full and truncated.
Pyramid as a three-dimensional figure
Everyone knows about Egyptian pyramids, therefore, it is well represented what figure will be discussed. Nevertheless, Egyptian stone structures are only a special case of a huge class of pyramids.
The geometric object under consideration 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 base plane. 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 the Euler equation:
2*n = (n+1) + (n+1) - 2.
The polygon located at the base gives the name of 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 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 there is an inclined pyramid.
A straight figure, the base of which is formed by an equilateral (equiangular) n-gon, is called regular.
Pyramid volume formula
To calculate the volume of the pyramid, we use the integral calculus. To do this, we divide the figure by secant 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 sectional layer is marked with a quadrilateral.
The area of each such layer can be calculated by the formula:
A(z) = A 0 *(h-z) 2 /h 2 .
Here A 0 is the area of the base, 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 arrive at the expression:
V = -A 0 *(h-z) 3 /(3*h 2)| h 0 \u003d 1/3 * A 0 * h.
We have obtained the formula for the volume of a pyramid. To find the value of V, it is enough to multiply the height of the figure by the area of \u200b\u200bthe 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
Received in paragraph above general formula for volume can be refined in the case of a pyramid with a regular base. The area of such a base is calculated by the following formula:
A 0 = n/4*L 2 *ctg(pi/n).
Here L is the length of the side regular polygon with n vertices. The symbol pi is the number pi.
Substituting the expression for A 0 into the general formula, we obtain the volume correct pyramid:
V n = 1/3*n/4*L 2 *h*ctg(pi/n) = n/12*L 2 *h*ctg(pi/n).
For example, for triangular pyramid this formula leads to the following expression:
V 3 \u003d 3/12 * L 2 * h * ctg (60 o) \u003d √3 / 12 * L 2 * h.
For a regular quadrangular pyramid, the volume formula takes the form:
V 4 \u003d 4/12 * L 2 * h * ctg (45 o) \u003d 1/3 * L 2 * h.
Determining the volumes of regular pyramids requires knowing the side of their base and the height of the figure.
Pyramid truncated
Suppose we have taken an arbitrary pyramid and cut off a part of its lateral surface containing the vertex. The remaining figure 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 is formed with parallel similar bases. 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 shows 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 an integral calculus similar to the above 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 pyramid of Cheops
It is curious to solve the problem of determining the volume that the largest Egyptian pyramid contains.
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 accuracy.
Let's use the given figures to determine the volume of this stone giant. Since the pyramid is a regular quadrangular, then the formula is valid for it:
Plugging in the numbers, we get:
V 4 \u003d 1/3 * (230.363) 2 * 146.5 ≈ 2591444 m 3.
The volume of the pyramid of Cheops 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!
- 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 the same isosceles trapezoids, whose area 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 the lateral surface area of a regular pyramid 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 the correct quadrangular pyramid. 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.
