Spatial geometry studies the properties of volumetric figures and their different relative positions. This article is devoted to the study of the characteristics of such a polyhedron as a pyramid. The inscribed and described figures in the cone and cube will be considered.
Spatial figure pyramid
A pyramid is a geometric figure bounded by n + 1 faces, of which one face is a polygon with n sides, and n faces represent triangles connected together at the same vertex.
The figure below shows two pyramids.
The left shape consists of 5 faces, where the polygon is a quadrangle, the right - of six faces, because its polygon has five sides. The quadrangle for the left figure and the pentagon for the right are the bases. The point where the triangles connect is the top of the pyramid.
As a geometric figure, a pyramid of arbitrary type can be obtained as follows: it is necessary to take an n-gon and connect all its angles with some fixed point in space, which, however, should not lie in the plane of the n-gon.
Every pyramid consists of n + 1 sides (faces), has 2 * n edges and n + 1 vertices.
What are the pyramids?
All types of pyramids in geometry differ from each other in two ways:
- type of polygonal base;
- the location of the top of the pyramid relative to the base.
Let's start with the relative position of the vertex. The perpendicular drawn to the base from it is called the height of the figure. If the height falls to the base exactly at the geometric center, then they say that the pyramid is straight. If the height falls to the base at any other point, then the figure is inclined.
In accordance with the type of n-gon, convex and concave pyramids are distinguished. In addition, the polygon gives the name of the whole figure. For example, a triangular base indicates that the pyramid itself is triangular, if the base is quadrangular, then the figure is called a quadrangular pyramid and so on.
A special case for solving many practical problems are regular pyramids. In geometry, they mean pyramids that have a regular polygon at the base and are straight. The set of regular pyramids with different polygons at the base is shown in the figure below.
Further in the article we will consider only such regular figures when studying the inscribed and described pyramids.
Regular Polygons and Circle
It is important to consider these flat figures in order to understand the theme of the inscribed and described pyramids. Let's start with the simplest of them - an equilateral triangle.
An equilateral triangle has 3 identical sides and three angles of 60 o . Its geometric center (barycenter) is located at the intersection of medians, which is also the intersection of heights and bisectors. If the side length of the triangle is a, then the circle circumscribed around it will go through all its vertices. Its center will be the barycenter of the triangle, and the radius will be equal to:
R 3 c = √3 / 3 * a
The inscribed circle will touch all sides of the triangle. Its center will be at the same point as for the circumscribed circle. The radius of the inscribed circle will be equal to:
R 3 i = √3 / 6 * a
Now we give similar formulas for a regular quadrangle, that is, for a square. It is not difficult to show that the radii of the circles described and inscribed in the square will be equal to:
R 4 c = a / √2;
R 4 i = a / 2
Where a is the length of the side of the square.
We write the formulas for a regular hexagon:
R 6 c = a;
R 6 i = a * √3 / 2
Here a is the side length of a regular hexagon.
All the above formulas are useful when considering the inscribed and described pyramids with respect to the cone.
Regular Polygons and Square
Before considering pyramids inscribed in a cube, appropriate formulas should be given for the lengths of the sides of the base of these figures inscribed in a square. Here we consider only two cases of a quadrangular pyramid.
In the first case, everything is simple, the lengths of the sides of the cube and the sides of the square base are equal, that is:
a i = l
Here a i is the side of the base of the pyramid, l is the length of the side of the cube.
In the second case, the regular quadrangular pyramid can be inscribed differently in a cube: the vertices of its base should be located in the middle of the sides of the square. Then we get the following formula for a i :
a i = l / √2
Pyramids inscribed in a cone
A cone is a three-dimensional figure, which at its base contains a circle. In fact, if you increase the number of sides of the n-coal base of the pyramid to infinity, then it will go into a cone.
The pyramid inscribed in the cone is located completely in its volume, that is, it does not extend beyond the base of the cone and its conical surface. Such a pyramid has common points with a cone at the base and at the top.
If the radius of the base of the cone is known, then the above formulas for determining the sides of regular polygons (triangle, square and hexagon) inscribed in a circle allow us to calculate the length of the side of the base of the pyramid. For example, the sides of a figure with a regular triangular base through the radius of the base of the cone R 3 c can be written as follows:
a = √3 * R 3 c
Knowing the side of the base and the height of the pyramid h, you can determine any of its characteristics. For example, the volume is calculated by the formula:
V = 1/3 * S 3 o * h
Where S 3 o - the area of an equilateral triangle with side a. The same formula is true for the volume of the cone, but instead of the area of the polygonal base, you should take the area of the circle on which the conical surface rests.
Circumscribed around the pyramid cone
In this case, we have a situation that is opposite to the previous one. Now the pyramid completely encloses a cone inside itself. The latter has a base radius, which is associated with the length of the side of the pyramid with the given formulas for the circle inscribed in the polygon.
For example, if a cone should be placed in a hexagonal pyramid with side a so that its base touches all sides of the hexagon, then it is necessary to take the radius R 6 i of the base of the cone, which will be equal to:
R 6 i = a * √3 / 2
Note that the height of both the described cone and inscribed in the pyramid is always equal to that for the latter.
Cube and pyramid
A cube is a regular polyhedron belonging to the class of prisms. The high symmetry of this figure allows you to enter different regular pyramids into it. The easiest way to enter it is the quadrangular pyramids.
In the simplest case of an inscribed pyramid, its base is one of the sides of the cube. The top of the pyramid will lie on the opposite parallel face of the cube in the center of the square.
The second variant of the arrangement of the pyramid inside the cube is as follows: if you connect the middle of one of the sides of the cube with each other, you will get a new square of a smaller size. He will be the base of the pyramid. Its top, as in the previous case, will be located in the middle of the opposite face of the cube.
Problem with a cone and a pyramid
Suppose that there is a cone described pyramid. The radius of the cone is 10 cm. It is necessary to calculate the volume of the pyramid, if it is known that the cone has a height of 15 cm, and the base of the pyramid is a regular triangle.
To calculate the side of the triangle, we use the corresponding formula:
a = √3 * R 3 c = √3 * 10 ≈ 17.32 cm
To determine the volume of the pyramid, calculate the area of its base. It is equal to:
S 3 o = √3 / 4 * a 2 = √3 / 4 * 17.32 2 ≈ 129.90 cm 2
Given that the cone is described near the pyramid, then the heights of these figures are the same. We substitute the corresponding value in the formula for the volume of the pyramid, we get the answer to the problem:
V = 1/3 * S 3 o * h = 1/3 * 129.90 * 15 = 649.5 cm 3
Task with a cube and a pyramid
There is a cube with side a into which a regular quadrangular pyramid is inscribed. It is necessary to calculate the ratio of the volume of the pyramid to that for the cube and find out whether the value obtained depends on the length of the side a.
Since the task does not say how exactly the pyramid is inscribed, two cases should be considered.
In the first case, we have a pyramid whose base length is a, its height is also equal to a. Then its volume will be:
V 1 = 1/3 * a 3
Its relation to the volume V c of the cube is equal to:
V 1 / V c = 1/3 * a 3 / a 3 = 1/3
In the second case of the inscribed pyramid in the cube, the side of its base will be equal to a / √2. The height of the figure remains the same. Then its volume is calculated as follows:
V 2 = 1/3 * a * (a / √2) 2 = 1/6 * a 3
And his relation to V c will be equal to:
V 2 / V c = 1/6 * a 3 / a 3 = 1/6
Thus, we have obtained that the ratio of the volumes of the quadrangular pyramid that is inscribed in the cube and the cube does not depend on the length of the edge of the latter.