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 three-dimensional 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 a plane on which the ends of its generators lie. Its formation occurs in the process of rotation of a right-angled triangle around any of its legs, therefore it belongs to the bodies of revolution.
Cone components
The following types of cones are distinguished: oblique (or oblique) and straight. Oblique is the one whose axis intersects with the center of its base not 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 its planebase at 90°.
That cone, the axis of which is perpendicular to its base, is called a straight cone. The axis and height in such a geometric body coincide due to the fact that the vertex in it is located above the center of the base diameter.
The cone consists of the following elements:
- The circle that is its base.
- Side.
- A point not lying in the plane of the base, called the top of the cone.
- Segments that connect the points of the circle of the base of the geometric body and its top.
All these segments are generatrices of the cone. They are inclined to the base of the geometric body, and in the case of a right cone their projections are equal, since the vertex is equidistant 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 base.
Since in an oblique (or inclined) body of revolution the vertex is displaced 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 be different.
The length of the generators in a right cone
As written earlier, 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, you can calculate the length of the generatrix, which will be equal to the sum of the squares of the base radius and height:
l2 =r2+ h2 or l=√r 2 + h2
where l is a generatrix;
r – radius;
h – height.
Generative in an oblique cone
Based on the fact that the generators in an oblique or oblique cone do not have the same length, it will not be possible to calculate them without additional constructions and calculations.
First of all, you need to know the height, the length of the axis and the radius of the base.
Having these data, you can calculate the part of the radius lying between the axis and the height, using the formula from the Pythagorean theorem:
r1=√k2 - h2
where r1 is the part of the radius between the axis and the height;
k – axle length;
h – height.
As a result of adding the radius (r) and its part lying between the axis and the height (r1), you can find out the full side of the right triangle formed by the generatrix of the cone, its height and diameter part:
R=r + r1
where R is the leg of the triangle formed by the height, generatrix and part of the diameter of the base;
r – base radius;
r1 – part of the radius between the axis and the height.
Using the same formula from the Pythagorean theorem, you can find the length of the generatrix of the cone:
l=√h2+ R2
or, without calculating R separately, combine the two formulas into one:
l=√h2 + (r + r1)2.
Despite whether it is a straight or oblique cone and what kind of input data, all methods for finding the length of the generatrix always come down to one result - the use of the Pythagorean theorem.
Cone section
Axial section of a cone is a plane passing along its axis or height. In a right cone, such a section is an isosceles triangle, in which the height of the triangle is the height of the body, its sides are the 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 a straight cone is the plane of its symmetry. The reason for this is that its top is 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 match in an inclined solid, the plane of the axial section may not include the height. If it is possible to construct a set of axial sections in such a cone, since only one condition must be observed for this - it must pass only through the axis, then only one axial section of the plane, which will belong to the height of this cone, can be drawn, because the number of conditions increases, and, as is known, two lines (together) can belong toonly one plane.
Section area
The axial section of the cone mentioned earlier is a triangle. Based on this, its area can be calculated using the formula for the area of a triangle:
S=1/2dh or S=1/22rh
where S is the cross-sectional area;
d – base diameter;
r – radius;
h – height.
In an oblique or oblique cone, the section along the axis is also a triangle, so the cross-sectional area in it is calculated similarly.
Volume
Since the cone is a three-dimensional figure in three-dimensional space, it is possible to calculate its volume. The volume of a cone is a number that characterizes this body in a volume unit, that is, in m3. The calculation does not depend on whether it is straight or oblique (oblique), since the formulas for these two types of bodies do not differ.
As stated earlier, the formation of a right 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 base plane of the body. However, such differences in structure do not affect the method of calculating its volume.
Volume calculation
The formula for the volume of any cone looks like this:
V=1/3πhr2
where V is the volume of the cone;
h – height;
r – radius;
π - constant equal to 3, 14.
In order to calculate the volume of a cone, you need to have data on the height and radius of the base of the body.
To calculate the height of a 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 triangle, the height can be calculated using the formula from the Pythagorean theorem (a2+ b2=c 2 or in our case h2+ r2=l2, where l - generatrix). In this case, the height will be calculated by extracting the square root of the difference between the squares of the hypotenuse and the other leg:
a=√c2- b2
That is, the height of the cone will be equal to the value obtained after extracting the square root from the difference between the square of the length of the generatrix and the square of the radius of the base:
h=√l2 - r2
Calculating the height using this method and knowing the radius of its base, you can calculate the volume of the cone. In this case, the generatrix plays an important role, since it serves as an auxiliary element in the calculations.
Similarly, if you know the height of the body and the length of its generatrix, you can find 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=√l2 - h2
Then, using the same formula as above, calculate the volume of the cone.
Inclined cone volume
Since the formula for the volume of a cone is the same for all types of a body of revolution, the difference in its calculation is the search for height.
In order to find out the height of an inclined cone, the input data must include the length of the generatrix, the radius of the base and the distance between the centerbase 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 base diameter, which will be the base of a right-angled 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.
