One of the interesting tasks that allow you to compare various three-dimensional figures are the tasks of describing one of them near the other. In this article, we will consider various options for the cylinder described near the pyramid and inscribed in the pyramid.
Pyramid in geometry
Before studying combinations of a cylinder inscribed in a pyramid and a pyramid inscribed in a cylinder, these figures should be considered from the point of view of geometry. Let's start with the pyramid.
The pyramid figure is a body in space, which is obtained by connecting all the vertices of an arbitrary flat n-gon with some point in space. Moreover, the n-gon can be completely arbitrary (convex, concave, regular, with a different number of sides n). The only condition imposed on the position of the marked point is that it must not lie in the plane in which the n-gon is located.
The picture above shows perhaps the most famous pyramid - a quadrangular. It can be seen that the vertices of the quadrangle, which is called the base of the figure, are connected to the point lying above it. This point is called the top of the pyramid.
The above definition and also the presented figure indicate that any pyramid, regardless of the type of its base, will include n triangles. All of them are connected at the top of the figure.
The perpendicular segment drawn from the top of the figure to its base is called height. If the height intersects the n-gon in the geometric center, then such a pyramid will be straight. Otherwise, there is an inclined figure.
If all sides of the n-gon are equal to each other, and the figure is straight, then it is called correct. It is with the right pyramids that it is convenient to work with when studying their mutual arrangement with other volume bodies in geometry.
Cylinder in geometry
In the general case, a cylinder can be obtained by moving a segment along a closed curve parallel to itself so that the segment does not lie in the plane of this curve. This segment is called the generatrix of the cylinder, and the curve along which it moves is called the guide.
If the guide is a circle, and the generatrix is perpendicular to it, then the resulting cylinder will be called straight with a round base. This figure is known to everyone. It is presented in the figure below.
Further we will consider only a straight round cylinder.
Unlike a pyramid, a cylinder has no vertices and edges. However, it is formed by two bases (two identical circles located in parallel planes) and a lateral cylindrical surface. If you look at the scan of this figure, you can see that it consists of two circles and one rectangle (see. Fig. Below).
The main characteristics of the cylinder are as follows:
- base radius;
- height - the distance between the bases;
- the area of the bases and lateral surface;
- volume of the figure.
Polygon and Circle
The last question that should be studied before considering the cylinder inscribed in the pyramid and described near it is related to the relative position of the regular polygon and the circle.
There are only two options for arranging these flat shapes:
- circle description of the n-gon;
- description of the n-gon of the circle.
We give formulas that allow us to calculate the length of the side of a polygon in terms of the radius of a circle. For example, consider only the first two polygons, that is, an equilateral triangle and a square.
If the circle passes through all the vertices of the n-gon, then they say that it describes it. With a known radius R, the side length is calculated by the formula:
for a triangle: a = √3 * R;
for the square: a = √2 * R
That is, the side of the square inscribed in a circle with radius R will be slightly smaller than that for an equilateral triangle described by the same circle.
If the circle touches each side of the n-gon, then they say that it is inscribed in it. In the case of regular polygons, the point of contact of the figures is exactly in the middle of each side of the n-gon. If the radius r of the inscribed circle is known, then the side of the n-gon is determined by the formula:
for a triangle: a = 2 * √3 * r;
for a square: a = 2 * r
That is, a triangle with a longer side length than a square can be described around a circle of a fixed radius.
Triangular pyramid inscribed in a cylinder
First, consider a simpler option, that is, when the pyramid is inside the cylinder. Let's analyze a concrete example with a regular triangular pyramid. Suppose that the radius R of the cylinder and its height h are known. It is necessary to find the characteristics of a regular triangular pyramid inscribed in a cylinder.
The formula for the side of an equilateral triangle inside the circle has already been given above. The length of its side is the length of the base of the pyramid. It is equal to:
a = √3 * R
The top of the inscribed pyramid lies exactly in the center of the upper base of the cylinder, so the heights of both figures are equal.
Knowing the length of the base side and the height of the regular triangular pyramid, one can calculate its other characteristics. For example, the volume is calculated by the formula:
V = √3 / 12 * a 2 * h
The length of the side rib a b can be calculated as follows:
a b = √ (R 2 + h 2 )
Quadrangular pyramid inscribed in a cylinder
As in the previous case, the pyramid is inside the cylinder. Only now its base is a square whose side through the radius R of the cylinder is calculated as follows:
a = √2 * R
The height of the pyramid is equal to that of the cylinder, that is, h.
The volume of the regular quadrangular pyramid (inscribed in the cylinder) is equal to:
V = 1/3 * a 2 * h
The length of the side rib a b is:
a b = √ (R 2 + h 2 )
Note that the formula for the length of the side rib is exactly the same as in the case of a triangular pyramid.
The cylinder is inscribed in the figure
A cylinder inscribed in a pyramid represents a more complex case of the location of these figures. To calculate the size of the pyramid according to the known radius and height of the cylinder, you should figure out how this cylinder will be located inside it.
Suppose there is a plane parallel to the base of the pyramid. Cross this side surface of the figure. The formed section will represent exactly the same polygon that lies at the base, but smaller. This polygon will describe the top base of the cylinder. The lower base will lie at the base of the pyramid.
To find the length of the side of the section polygon, one should use the function of the dependence of the section area on the vertical coordinate z. This function has the form:
S (z) = (h p -z) 2 / h p 2 * S 0
Here z is the distance from the base of the pyramid along its height, h p is the height of the pyramid.
How to use this formula to determine the parameters of the pyramid described near the cylinder, we show on the example of solving the problem.
The task with a quadrangular pyramid and a cylinder
It is known that the cylinder has a radius of r = 5 cm and a height of h = 6 cm. Find the height and side of the regular quadrangular pyramid that describes it.
The upper base of the cylinder should fit into a square slice at a height of h = 6 cm from the base of the pyramid. Then the cross-sectional area is equal to:
S (6) = (h p -6) 2 / h p 2 * a 2
Here a is the side of the base of the pyramid. If we take the square root of S (6), then we obtain the side length of the square section. It must be equal to 2 * r, so that the base of the cylinder can fit into this section, then we get:
√S (6) = (h p -6) / h p * a = 2 * r = 10
From here we get the expression:
a = 10 * h p / (h p -6)
Thus, it is possible to enter the cylinder defined by the condition of the problem not into one single regular quadrangular pyramid, but into an infinite number of them. However, the parameters of each of them must satisfy the expression above, which relates the height of the figure to the length of the side of its base.