The ability to calculate the volume of spatial figures is important in solving a number of practical problems in geometry. One of the common shapes is the pyramid. In this article, we consider the formula for the volume of the pyramid, both complete and truncated.
Pyramid as a three-dimensional figure
Everyone knows about the Egyptian pyramids, therefore, it is well aware 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 the 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 in question is a perfect polyhedron, the numbers of the marked elements obey Euler's equality:
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 connect 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 whose base is formed by an equilateral (equilateral) 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 parallel to the base by secant planes into an infinite number of thin layers. The figure below shows a quadrangular pyramid of 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 by the formula:
A (z) = A 0 * (hz) 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 * (hz) 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 expression obtained is valid for calculating the volume of a pyramid of arbitrary type. That is, it can be inclined, and its base can be an arbitrary n-gon.
The correct pyramid and its volume
The general formula for volume obtained in paragraph above can be clarified in the case of a pyramid with the correct 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 side length of a 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 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 a regular 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 knowledge of the side of their base and the height of the figure.
Truncated pyramid
Suppose we took an arbitrary pyramid and cut off part of the side surface containing the vertex from it. The remaining shape is called a truncated pyramid. It already consists of two n-coal bases and n trapezoids that connect them. If the secant 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 a certain coefficient k.
The figure above shows the truncated regular hexagonal pyramid. It can be seen that its upper base, like the lower one, is formed by a regular hexagon.
The formula for the volume of a truncated pyramid, which can be derived using an integral calculus similar to the above, has the form:
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 interesting to solve the problem of determining the volume that the largest Egyptian pyramid encloses within itself.
In 1984, British Egyptologists Mark Lehner and John Goodman established the exact dimensions of the Cheops pyramid. Its initial height was 14.60 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 figures given to determine the volume of this stone giant. Since the pyramid is regular quadrangular, then the formula is valid for it:
V 4 = 1/3 * L 2 * h.
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 pyramid of Cheops will need more than 1000 of these pools!