Geometry is a branch of mathematics that studies structures in space and the relationship between them. In turn, it also consists of sections, and one of them is stereometry. It provides for the study of the properties of volumetric figures located in space: a cube, a pyramid, a ball, a cone, a cylinder, etc.
A cone is a body in Euclidean space that bounds a conical surface and the plane on which the ends of its generators lie. Its formation occurs during the rotation of a rectangular triangle around any of its legs, therefore it refers to bodies of revolution.
Cone components
The following types of cones are distinguished: oblique (or inclined) and straight. An oblique is one whose axis does not intersect with the center of its base at a right angle. For this reason, the height in such a cone does not coincide with the axis, since it is a segment that is lowered from the top of the body to the plane of its base at an angle of 90 Β°.
That cone whose axis is perpendicular to its base is called straight. The axis and height in such a geometric body coincide because the peak in it is located above the center of the diameter of the base.
The cone consists of the following elements:
- The circle that is its foundation.
- Side surface.
- A point that does not lie in the plane of the base, called the vertex of the cone.
- The line segments that connect the points of the circle base of the geometric body and its peak.
All these segments are generators of the cone. They are inclined to the base of the geometric body, and in the case of a straight cone their projections are equal, since the vertex is equally spaced from the points of the base circle. Thus, we can conclude that in a regular (straight) cone, the generators are equal, that is, they have the same length and form the same angles with the axis (or height) and the base.
Since in the oblique (or inclined) body of revolution, the vertex is offset relative to the center of the base plane, the generators in such a body have different lengths and projections, since each of them is at a different distance from any two points of the base circle. In addition, the angles between them and the height of the cone will also differ.
The length of the generators in a straight cone
As previously written, the height in a straight geometric body of revolution is perpendicular to the plane of the base. Thus, the generatrix, height and radius of the base create a right triangle in the cone.
That is, knowing the radius of the base and the height, using the formula from the Pythagorean theorem, we can calculate the length of the generatrix, which will be equal to the sum of the squares of the radius of the base and height:
l 2 = r 2 + h 2 or l = βr 2 + h 2
where l is the generatrix;
r is the radius;
h is the height.
Forming in an inclined cone
Based on the fact that the generators in the oblique or inclined cone are not of the same length, they cannot be calculated without additional constructions and calculations.
First of all, you need to know the height, length of the axis and radius of the base.
Having these data, it is possible to calculate the part of the radius lying between the axis and the height using the formula from the Pythagorean theorem:
r 1 = βk 2 - h 2
where r 1 is the part of the radius between the axis and the height;
k is the length of the axis;
h is the height.
As a result of the addition of the radius (r) and its part lying between the axis and the height (r 1 ), we can find out the full side of the right-angled triangle formed by the generatrix of the cone, its height and part of the diameter:
R = r + r 1
where R is the leg of the triangle formed by the height, forming and part of the diameter of the base;
r is the radius of the base;
r 1 is the part of the radius between the axis and the height.
Using all the same formula from the Pythagorean theorem, we can find the length of the generatrix of the cone:
l = βh 2 + R 2
or, without separately calculating R, combine the two formulas into one:
l = βh 2 + (r + r 1 ) 2 .
Despite the fact that there is a straight or oblique cone and what input data, all methods of finding the length of the generatrix always come down to one result - using the Pythagorean theorem.
Cone section
The axial section of a cone is a plane passing along its axis or height. In a straight cone, such a section is an isosceles triangle in which the height of the triangle is the height of the body, its sides are generators, and the base is the diameter of the base. In an equilateral geometric body, the axial section is an equilateral triangle, since in this cone the diameter of the base and the generators are equal.
The plane of the axial section in the straight cone is the plane of its symmetry. The reason for this is that its top is located above the center of its base, that is, the plane of the axial section divides the cone into two identical parts.
Since the height and axis do not coincide in an inclined volumetric body, the plane of the axial section may not include height. If there are many axial sections in such a cone, since for this it is necessary to observe only one condition - it must pass only through the axis, then the axial section of the plane to which the height of this cone belongs can only be drawn, because the number of conditions increases, and, as you know, two lines (together) can belong to only one plane.
Cross-sectional area
The axial section of the cone mentioned earlier is a triangle. Based on this, its area can be calculated by the area of ββthe triangle:
S = 1/2 * d * h or S = 1/2 * 2r * h
where S is the cross-sectional area;
d is the diameter of the base;
r is the radius;
h is the height.
In an oblique or inclined cone, the section along the axis is also a triangle, therefore, the section area in it is calculated in the same way.
Volume
Since the cone is a three-dimensional figure in three-dimensional space, its volume can be calculated. The volume of a cone is the number that characterizes this body in a unit of measurement of volume, that is, in m 3 . The calculation does not depend on whether it is straight or oblique (inclined), since the formulas for these two types of bodies do not differ.
As indicated earlier, the formation of a straight cone occurs due to the rotation of a right triangle along one of its legs. An inclined or oblique cone is formed differently, since its height is shifted away from the center of the plane of the base of the body. Nevertheless, such differences in structure do not affect the methodology for calculating its volume.
Volume calculation
The formula for the volume of any cone is as follows:
V = 1/3 * Ο * h * r 2
where V is the volume of the cone;
h is the height;
r is the radius;
Ο is a constant equal to 3.14.
In order to calculate the volume of the cone, it is necessary to have data on the height and radius of the base of the body.
To calculate the height of the body, you need to know the radius of the base and the length of its generatrix. Since the radius, height and generatrix are combined into a right-angled triangle, the height can be calculated using the formula from the Pythagorean theorem (a 2 + b 2 = c 2 or in our case h 2 + r 2 = l 2 , where l is the generatrix). The height will be calculated by extracting the square root of the difference between the squares of the hypotenuse and another leg:
a = βc 2 - b 2
That is, the height of the cone will be equal to the value obtained after extracting the square root of the difference between the square of the length of the generatrix and the square of the radius of the base:
h = βl 2 - r 2
By calculating the height using this method and knowing the radius of its base, it is possible to calculate the volume of the cone. The generator plays an important role in this case, since it serves as an auxiliary element in the calculations.
Similarly, if the height of the body and the length of its generatrix are known, you can find out the radius of its base by extracting the square root of the difference between the square of the generatrix and the square of the height:
r = βl 2 - h 2
Then, using the same formula as above, calculate the volume of the cone.
Inclined cone volume
Since the formula for the volume of the cone is the same for all types of bodies of revolution, the difference in its calculation is the search for height.
In order to know the height of the inclined cone, the input data must include the length of the generatrix, the radius of the base and the distance between the center of the base and the intersection of the height of the body with the plane of its base. Knowing this, you can easily calculate that part of the diameter of the base, which will be the base of the right triangle (formed by the height, the generatrix and the plane of the base). Then, again using the Pythagorean theorem, calculate the height of the cone, and subsequently its volume.