What is the generator of the cone? How to find the generatrix of a cone
We know what a cone is, let's try to find its surface area. Why do you need to solve such a problem? For example, you need to understand how much dough will go into making a waffle cone? Or how many bricks does it take to make a brick castle roof?
Measuring the lateral surface area of a cone simply cannot be done. But let’s imagine the same horn wrapped in fabric. To find the area of a piece of fabric, you need to cut it and lay it out on the table. It will work out flat figure, we can find its area.
Rice. 1. Section of a cone along the generatrix
Let's do the same with the cone. Let's "cut" it lateral surface along any generatrix, for example (see Fig. 1).
Now let’s “unwind” the side surface onto a plane. We get a sector. The center of this sector is the vertex of the cone, the radius of the sector is equal to the generatrix of the cone, and the length of its arc coincides with the circumference of the base of the cone. This sector is called the development of the lateral surface of the cone (see Fig. 2).
Rice. 2. Development of the side surface
Rice. 3. Angle measurement in radians
Let's try to find the area of the sector using the available data. First, let's introduce the notation: let the angle at the vertex of the sector be in radians (see Fig. 3).
We will often have to deal with the angle at the top of the sweep in problems. For now, let’s try to answer the question: can’t this angle turn out to be more than 360 degrees? That is, wouldn’t it turn out that the sweep would overlap itself? Of course not. Let's prove this mathematically. Let the scan “superpose” on itself. This means that the sweep arc length longer circle radius . But, as already mentioned, the length of the sweep arc is the length of the circle of radius . And the radius of the base of the cone, of course, is less than the generatrix, for example, because the leg right triangle less than hypotenuse
Then let’s remember two formulas from the planimetry course: arc length. Sector area: .
In our case, the role is played by the generator , and the length of the arc is equal to the circumference of the base of the cone, that is. We have:
Finally we get: .
Along with the area of the lateral surface, one can also find the area full surface. To do this, add the area of the base to the area of the lateral surface. But the base is a circle of radius, whose area according to the formula is equal to .
Finally we have: , where is the radius of the base of the cylinder, is the generatrix.
Let's solve a couple of problems using the given formulas.
Rice. 4. Required angle
Example 1. The development of the lateral surface of the cone is a sector with an angle at the apex. Find this angle if the height of the cone is 4 cm and the radius of the base is 3 cm (see Fig. 4).
Rice. 5. Right Triangle Forming a Cone
By the first action, according to the Pythagorean theorem, we find the generator: 5 cm (see Fig. 5). Next, we know that .
Example 2. The axial cross-sectional area of the cone is equal to , the height is equal to . Find the total surface area (see Fig. 6).
Here are problems with cones, the condition is related to its surface area. In particular, in some problems there is a question of changing the area when increasing (decreasing) the height of the cone or the radius of its base. Theory for solving problems in . Let's consider the following tasks:
27135. The circumference of the base of the cone is 3, the generator is 2. Find the area of the lateral surface of the cone.
The lateral surface area of the cone is equal to:
Substituting the data:
75697. How many times will the area of the lateral surface of the cone increase if its generatrix is increased by 36 times, and the radius of the base remains the same?
Cone lateral surface area:
The generatrix increases 36 times. The radius remains the same, which means the circumference of the base has not changed.
This means that the lateral surface area of the modified cone will have the form:
Thus, it will increase by 36 times.
*The relationship is straightforward, so this problem can be easily solved orally.
27137. How many times will the area of the lateral surface of the cone decrease if the radius of its base is reduced by 1.5 times?
The lateral surface area of the cone is equal to:
The radius decreases by 1.5 times, that is:
It was found that the lateral surface area decreased by 1.5 times.
27159. The height of the cone is 6, the generatrix is 10. Find the area of its total surface divided by Pi.
Full cone surface:
You need to find the radius:
The height and generatrix are known, using the Pythagorean theorem we calculate the radius:
Thus:
Divide the result by Pi and write down the answer.
76299. The total surface area of the cone is 108. A section is drawn parallel to the base of the cone, dividing the height in half. Find the total surface area of the cut off cone.
The section passes through the middle of the height parallel to the base. This means that the radius of the base and the generatrix of the cut off cone will be 2 times less than the radius and generatrix of the original cone. Let us write down the surface area of the cut off cone:
Got it to be 4 times less area surface of the original, that is, 108:4 = 27.
*Since the original and cut off cone are similar bodies, it was also possible to use the similarity property:
27167. The radius of the base of the cone is 3 and the height is 4. Find the total surface area of the cone divided by Pi.
Formula for the total surface of a cone:
The radius is known, it is necessary to find the generatrix. 
According to the Pythagorean theorem:
Thus:
Divide the result by Pi and write down the answer.
Task. The area of the lateral surface of the cone is four times the area of the base. Find what is the cosine of the angle between the generatrix of the cone and the plane of the base.
The area of the base of the cone is: 
The bodies of rotation studied in school are the cylinder, cone and ball.
If in a problem on the Unified State Exam in mathematics you need to calculate the volume of a cone or the area of a sphere, consider yourself lucky.
Apply formulas for volume and surface area of a cylinder, cone and sphere. All of them are in our table. Learn by heart. This is where knowledge of stereometry begins.
Sometimes it's good to draw the view from above. Or, as in this problem, from below.
2. How many times is the volume of a cone circumscribed about a regular quadrangular pyramid greater than the volume of a cone inscribed in this pyramid?
It's simple - draw the view from below. We see that the radius of the larger circle is times larger than the radius of the smaller one. The heights of both cones are the same. Therefore, the volume of the larger cone will be twice as large.
Another important point. Remember that in the problems of part B Unified State Exam options in mathematics, the answer is written as a whole number or a finite decimal fraction. Therefore, there should not be any or in your answer in part B. There is no need to substitute the approximate value of the number either! It must definitely shrink! It is for this purpose that in some problems the task is formulated, for example, as follows: “Find the area of the lateral surface of the cylinder divided by.”
Where else are the formulas for volume and surface area of bodies of revolution used? Of course, in problem C2 (16). We will also tell you about it.
Geometry is a branch of mathematics that studies structures in space and the relationships between them. In turn, it also consists of sections, and one of them is stereometry. It involves the study of the properties of three-dimensional figures located in space: cube, pyramid, ball, cone, cylinder, etc.
A cone is a body in Euclidean space that is bounded by a conical surface and the plane on which the ends of its generators lie. Its formation occurs during the rotation of a right triangle around any of its legs, so it belongs to bodies of revolution.
Components of a cone
There are the following types of cones: oblique (or inclined) and straight. 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°.
The cone whose axis is perpendicular to its base is called straight. 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 diameter of the base.
The cone consists of the following elements:
- The circle that is its base.
- Lateral surface.
- A point not lying in the plane of the base, called the vertex of the cone.
- Segments that connect the points of the circle of the base of a geometric body and its vertex.
All these segments are generators 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 circle of the base. 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 an oblique (or inclined) body of revolution the vertex is shifted relative to the center of the base plane, the generatrices in such a body have different lengths and projection, since each of them is at a different distance from any two points on the circle of the base. In addition, the angles between them and the height of the cone will also be different.
Length of generatrices in a straight cone
As written earlier, the height in a right geometric body of rotation 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 base radius and 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:
l 2 = r 2 + h 2 or l = √r 2 + h 2
where l is the generator;
r - radius;
h - height.
Generator in an inclined cone
Based on the fact that in an oblique or inclined cone the generators 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, axis length and base radius.
r 1 = √k 2 - h 2
where r 1 is the part of the radius between the axis and the height;
k - axis length;
h - height.
As a result of adding the radius (r) and its part lying between the axis and the height (r 1), you can find out the complete generated generatrix of the cone, its height and part of the diameter:
where R is the leg of a triangle formed by the height, the generator and part of the diameter of the base;
r - radius of the base;
r 1 - 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 = √h 2 + R 2
or, without separately calculating R, combine the two formulas into one:
l = √h 2 + (r + r 1) 2.
Regardless of whether the cone is straight or oblique and what the input data are, all methods for finding the length of the generatrix always come down to one result - the use of the Pythagorean theorem.
Cone section
Axial is a plane passing along its axis or height. In a straight cone, such a section is 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 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 axial section plane may not include the height. If many axial sections in such a cone can be constructed, since for this only one condition must be met - it must pass only through the axis, then the axial section of the plane to which the height of this cone will belong can be drawn only one, because the number of conditions increases, and, as is known, two straight lines (together) can belong to only one plane.
Cross-sectional area
The previously mentioned axial section of the cone is a triangle. Based on this, its area can be calculated using the formula for the area of a triangle:
S = 1/2 * d * h or S = 1/2 * 2r * h
where S is the cross-sectional area;
d - base diameter;
r - radius;
h - height.
In an oblique or inclined cone, the cross-section along the axis is also a triangle, so the cross-sectional area in it is calculated in a similar way.
Volume
Since the cone is voluminous figure in three-dimensional space, then its volume can be calculated. The volume of a cone is a number that characterizes this body in a unit of volume, 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 plane of the base of the body. Nevertheless, such differences in structure do not affect the method for calculating its volume.
Volume calculation
Any cone looks like this:
V = 1/3 * π * h * r 2
where V is the volume of the cone;
h - height;
r - radius;
π is a constant equal to 3.14.
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 generator are combined into a right 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 generator). The height will be calculated by taking the square root of the difference between the squares of the hypotenuse and the other leg:
a = √c 2 - b 2
That is, the height of the cone will be equal to the value obtained after taking 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 = √l 2 - r 2
By calculating the height using this method and knowing the radius of its base, you can calculate the volume of the cone. The teacher plays important role, since it serves auxiliary element in calculations.
Similarly, if the height of a body and the length of its generatrix are known, one can find out the radius of its base by extracting Square root from the difference between the square of the generator and the square of the height:
r = √l 2 - h 2
Then, using the same formula as above, calculate the volume of the cone.
Volume of an inclined cone
Since the formula for the volume of a cone is the same for all types of bodies of rotation, 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 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 base diameter that will be the base of a 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.