Every student has heard of a round cone and imagines what this three-dimensional figure looks like. This article defines the development of a cone, provides formulas that describe its characteristics, and describes how to construct it using a compass, protractor and straightedge.
Circular cone in geometry
Let's give a geometric definition of this figure. A round cone is a surface that is formed by straight line segments connecting all points of a certain circle with a single point in space. This single point must not belong to the plane in which the circle lies. If we take a circle instead of a circle, then this method also leads to a cone.
The circle is called the base of the figure, its circumference is the directrix. The segments connecting the point with the directrix are called generatrices or generators, and the point where they intersect is the vertex of the cone.
Round cone can be straight and oblique. Both figures are shown in the figure below.
The difference between them is this: if the perpendicular from the top of the cone falls exactly to the center of the circle, then the cone will be straight. For him, the perpendicular, which is called the height of the figure, is part of his axis. In the case of an oblique cone, the height and axis form an acute angle.
Due to the simplicity and symmetry of the figure, we will further consider the properties of only a right cone with a round base.
Getting a shape using rotation
Before proceeding to consider the development of the surface of a cone, it is useful to know how this spatial figure can be obtained using rotation.
Suppose we have a right triangle with sides a, b, c. The first two of them are legs, c is the hypotenuse. Let's put a triangle on leg a and start to rotate it around leg b. The hypotenuse c will then describe a conical surface. This simple cone technique is shown in the diagram below.
Obviously, leg a will be the radius of the base of the figure, leg b will be its height, and the hypotenuse c corresponds to the generatrix of a round right cone.
View of the development of the cone
As you might guess, the cone is formed by two types of surfaces. One of them is a flat base circle. Suppose it has radius r. The second surface is lateral and is called conical. Let its generator be equal to g.
If we have a paper cone, then we can take scissors and cut off the base from it. Then, the conical surface should be cutalong any generatrix and deploy it on the plane. In this way, we obtained a development of the lateral surface of the cone. The two surfaces, together with the original cone, are shown in the diagram below.
The base circle is depicted at the bottom right. The unfolded conical surface is shown in the center. It turns out that it corresponds to some circular sector of the circle, the radius of which is equal to the length of the generatrix g.
Angle and area sweep
Now we get formulas that, using the known parameters g and r, allow us to calculate the area and angle of the cone.
Obviously, the arc of the circular sector shown above in the figure has a length equal to the circumference of the base, that is:
l=2pir.
If the whole circle with radius g were built, then its length would be:
L=2pig.
Since the length L corresponds to 2pi radians, then the angle on which the arc l rests can be determined from the corresponding proportion:
L==>2pi;
l==> φ.
Then the unknown angle φ will be equal to:
φ=2pil/L.
Substituting the expressions for the lengths l and L, we arrive at the formula for the angle of development of the lateral surface of the cone:
φ=2pir/g.
The angle φ here is expressed in radians.
To determine the area Sbof a circular sector, we will use the found value of φ. We make one more proportion, only for the areas. We have:
2pi==>pig2;
φ==> Sb.
From where to express Sb, and then substitute the value of the angle φ. We get:
Sb=φg2pi/(2pi)=2pir/gg 2/2=pirg.
For the area of a conical surface, we have obtained a fairly compact formula. The value of Sb is equal to the product of three factors: pi, the radius of the figure and its generatrix.
Then the area of the entire surface of the figure will be equal to the sum of Sb and So (circular base area). We get the formula:
S=Sb+ So=pir(g + r).
Building a sweep of a cone on paper
To complete this task you will need a piece of paper, a pencil, a protractor, a ruler and a compass.
First of all, let's draw a right-angled triangle with sides 3 cm, 4 cm and 5 cm. Its rotation around the leg of 3 cm will give the desired cone. The figure has r=3 cm, h=4 cm, g=5 cm.
Building a sweep will start by drawing a circle with radius r with a compass. Its length will be equal to 6pi cm. Now next to it we will draw another circle, but with radius g. Its length will correspond to 10pi cm. Now we need to cut off a circular sector from a large circle. Its angle φ is:
φ=2pir/g=2pi3/5=216o.
Now we set aside this angle with a protractor on a circle with radius g and draw two radii that will limit the circular sector.
SoThus, we have built a development of the cone with the specified parameters of radius, height and generatrix.
An example of solving a geometric problem
Given a round straight cone. It is known that the angle of its lateral sweep is 120o. It is necessary to find the radius and generatrix of this figure, if it is known that the height h of the cone is 10 cm.
The task is not difficult if we remember that a round cone is a figure of rotation of a right triangle. From this triangle follows an unambiguous relationship between height, radius and generatrix. Let's write the corresponding formula:
g2=h2+ r2.
The second expression to use when solving is the formula for the angle φ:
φ=2pir/g.
Thus, we have two equations relating two unknown quantities (r and g).
Express g from the second formula and substitute the result into the first, we get:
g=2pir/φ;
h2+ r2=4pi2r 2/φ2=>
r=h /√(4pi2/φ2 - 1).
Angle φ=120o in radians is 2pi/3. We substitute this value, we get the final formulas for r and g:
r=h /√8;
g=3h /√8.
It remains to substitute the height value and get the answer to the problem question: r ≈ 3.54 cm, g ≈ 10.61 cm.
