The ability to calculate the volume of spatial figures is important in solving a number of practical problems in geometry. One of the most common shapes is the pyramid. In this article, we will consider the formulas for the volume of the pyramid, both full and truncated.
Pyramid as a three-dimensional figure
Everyone knows about the Egyptian pyramids, so they have a good idea of what figure will be discussed. However, Egyptian stone structures are only a special case of a huge class of pyramids.
The considered geometrical 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 base plane. This definition leads to a figure consisting of one n-gon and n triangles.
Any pyramid consists of n+1 faces, 2n edges and n+1 vertices. Since the figure under consideration is a perfect polyhedron, the numbers of marked elements obey the Euler equation:
2n=(n+1) + (n+1) - 2.
The polygon 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 whose base 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 layer of section is marked with a quadrilateral.
The area of each such layer can be calculated using the formula:
A(z)=A0(h-z)2/h2.
Here A0 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 A0.
To get the formula for the volume of a pyramid, you should calculate the integral over the entire height of the figure, that is:
V=∫h0(A(z)dz).
Substituting the dependence A(z) and calculating the antiderivative, we arrive at the expression:
V=-A0(h-z)3/(3h2)| h0=1/3A0h.
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 \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 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 the paragraph above can be refined in the case of a pyramid with the correct base. The area of such a base is calculated using the following formula:
A0=n/4L2ctg(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 A0 into the general formula, we get the volume of a regular pyramid:
V=1/3n/4L2hctg(pi/n)=n/12 L2hctg(pi/n).
For example, for a triangular pyramid, this formula leads to the following expression:
V3=3/12L2hctg(60o)=√3/12L2h.
For a regular quadrangular pyramid, the volume formula becomes:
V4=4/12L2hctg(45o)=1/3L2h.
Determining the volume of regular pyramids requires knowing the side of their base and the height of the figure.
Truncated pyramid
Suppose we tookan arbitrary pyramid and cut off a part of its lateral surface containing the top. 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 picture above shows a 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, is:
V=1/3h(A0+ A1+ √(A0 A1)).
Where A0 and A1 are the areas of the lower (large) and upper (small) bases, respectively. The variable h is the height of the truncated pyramid.
The volume of the pyramid of Cheops
It is interesting to solve the problem of determining the volume that the largest Egyptian pyramid contains inside.
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:
V4=1/3L2h.
Substitute the numbers, we get:
V4=1/3(230, 363)2146, 5 ≈ 2591444 m 3.
The volume of the Cheops pyramid is almost 2.6 million m3. For comparison, we note that the Olympic pool has a volume of 2.5 thousand m3. That is, to fill the entire Cheops pyramid, more than 1000 of these pools will be needed!
