When considering figures in space, problems often arise in determining their surface area. One such figure is a cone. Consider in the article what is the lateral surface of a cone with a round base, as well as a truncated cone.
Round base cone
Before proceeding to the consideration of the lateral surface of the cone, we show what kind of figure this is and how to obtain it by geometric methods.
Take a right triangle ABC, in which AB and AC are legs. We put this triangle on the side of AC and we will rotate it around side of AB. As a result, the sides AC and BC will describe the two surfaces of the figure, which is shown below.
The figure obtained by rotation is called a round straight cone. It is round because its base is a circle, and straight because a perpendicular drawn from the top of the figure (point B) intersects the circle in its center. The length of this perpendicular is called the height. Obviously, it is equal to leg AB. The height is usually denoted by the letter h.
In addition to height, the cone in question is described by two more linear characteristics:
- generatrix, or generatrix (hypotenuse BC);
- radius of the base (leg AC).
The radius is denoted by the letter r, and the generator by g. Then, taking into account the Pythagorean theorem, we can write down the equality important for the figure under consideration:
g 2 = h 2 + r 2
Conical surface
The set of all generatrices forms a conical or lateral surface of the cone. In appearance, it is difficult to say which flat figure it corresponds to. The latter is important to know when determining the area of a conical surface. To solve this problem, use the sweep method. It consists in the following: along an arbitrary generatrix mentally cut the surface, and then deploy it on a plane. With this method of obtaining a scan, the following flat figure is formed.
As you might guess, the circle corresponds to the base, but the circular sector is a conical surface, the area of which we are interested in. The sector is bounded by two generatrices and an arc. The length of the latter is exactly equal to the perimeter (length) of the circumference of the base. These characteristics uniquely determine all the properties of the circular sector. We will not give intermediate mathematical calculations, but write down immediately the final formula, using which we can calculate the area of the lateral surface of the cone. The formula is:
S b = pi * g * r
The area of the conical surface S b is equal to the product of two parameters and the number Pi.
Truncated cone and its surface
If we take an ordinary cone and cut off the top of it with a parallel plane, then the remaining figure will represent a truncated cone. Its lateral surface is bounded by two round bases. Denote their radii as R and r. The height of the figure is denoted by h, and by the generator, by g. The following is a paper scan for this shape.
It can be seen that the lateral surface is no longer a circular sector, it is smaller in area, since the central part was cut off from it. The scan is limited to four lines, two of them are straight segments-generatrixes, the other two are arcs with the lengths of the corresponding circumferences of the bases of the truncated cone.
The lateral surface S b is calculated as follows:
S b = pi * g * (r + R)
Generatrix, radii and height are related by the following equality:
g 2 = h 2 + (R - r) 2
The problem with the equality of the area of figures
Given a cone with a height of 20 cm and a radius of the base of 8 cm. It is necessary to find the height of the truncated cone, the side surface of which will have the same area as this cone. The truncated figure is built on the same base, and the radius of the upper base is 3 cm.
First of all, we write the condition for the equality of the areas of the cone and the truncated figure. We have:
S b1 = S b2 =>
pi * g 1 * R = pi * g 2 * (r + R)
Now we write the expressions for the generators of each figure:
g 1 = √ (R 2 + h 1 2 );
g 2 = √ ((Rr) 2 + h 2 2 )
Substitute g 1 and g 2 in the formula for equal areas and squared the left and right sides, we get:
R 2 * (R 2 + h 1 2 ) = ((Rr) 2 + h 2 2 ) * (r + R) 2
From where we get the expression for h 2 :
h 2 = √ (R 2 * (R 2 + h 1 2 ) / (r + R) 2 - (R - r) 2 )
We will not simplify this equality, but simply substitute the data known from the condition:
h 2 = √ (8 2 * (8 2 + 20 2 ) / (3 + 8) 2 - (8 - 3) 2 ) ≈ 14.85 cm
Thus, so that the area of the lateral surfaces of the figures is equal, the truncated cone should have the following parameters: R = 8 cm, r = 3 cm, h 2 ≈ 14.85 cm.