Each volumetric figure, which has finite linear dimensions, has a certain surface area in space. In the article, we consider what the area of the lateral surface of the cone is equal to, we present the corresponding formulas and show where they are derived from.
What is a cone?
In the general case, a cone in geometry is any spatial figure that is formed as a result of connecting a fixed point in space with all points of some plane curve. A fixed point is called the top of the shape. The segments connecting it with the curve are called generatrices, or generators, since their combination forms a conical surface. The curve on which this surface rests is called the directrix, that is, the guide. A directrix can be an arbitrary curve, for example, a hyperbola, a circle, a parabola, an ellipse, and so on. The cone formed on them will be hyperbolic, round, parabolic and elliptic, respectively.
The figure above shows an example of two identical elliptical cones facing each other with their vertices.
Round cone
The area of the lateral surface of the cone will be considered as an example of a round straight figure. Such a cone is a round base on which the conical surface rests. This figure is shown below.
All generators of this figure are equal to each other. Their length is always greater than the radius of the base. The distance from the top of the cone to its circular base is called height. The height intersects the circle in its center, so the cone is called straight.
Obtaining this cone is not difficult. To do this, take any triangle having a right angle, and rotate it around one of the legs as shown below in the diagram.
If the hypotenuse of this triangle is denoted by the letter g, and its legs are h and r, then the equality holds:
g 2 = h 2 + r 2 .
For the resulting cone, g is the generator, h is the height, r is the radius of the circle.
What is the lateral surface of a cone with a round base?
The answer to this question is easiest if you cut a conical surface along one of the generatrixes and deploy it on a plane. The resulting shape is called a side scan. It is shown in the main photo for the article, which also gives a circle - the base of the figure.
This scan shows that the area of the lateral surface of the cone is equal to the area of the corresponding circular sector. It is bounded by two generatrices g, which represent the radius of a full circle, and an arc. The length of the latter is exactly equal to the circumference of the base. We get the formula for the area of this sector.
First, we determine the angle in radians corresponding to the arc of the sector. It can be found using the following proportion:
2 * pi ==> 2 * pi * g;
x ==> 2 * pi * r.
Here 2 * pi * g is the length of the entire circle bounding the sector under consideration, 2 * pi * r is the length of the arc of the sector. The angle in radians x sectors will be equal to:
x = 2 * pi * r * 2 * pi / (2 * pi * g) = 2 * pi * r / g.
To determine the area of the sector under consideration, one should use the proportion through the corresponding areas. We have:
2 * pi ==> pi * g 2 ;
2 * pi * r / g ==> S b .
Here pi * g 2 is the area of the circle constructed using the generator g, S b is the area of the side surface of the cone equal to the area of the circular sector under consideration. The result of solving the proportion will be the final formula for S b :
S b = pi * g 2 * 2 * pi * r / g / (2 * pi) = pi * r * g.
Thus, to find the area of the conical surface, it is enough to multiply the radius of the figure by its directrix and by the number pi.
In obtaining the final formula for S b through proportions, we used the property of equality of the angle of the full circle to the number 2 * pi radians.
The concept of a truncated cone
Let there be a round straight cone. Take a plane and cut off the upper part from this figure so that the secant plane runs parallel to the base of the cone. The figure remaining under the plane is called a straight truncated cone with parallel bases. It is shown in the figure below.
Unlike the original figure, a truncated cone is formed by three surfaces:
- small round base;
- large round base;
- part of the conical surface.
The last in the list is the side surface for the figure in question.
For a truncated figure, the same concepts are valid as for a complete cone. So, the distance between its bases is the height h, each base has its own radius (r 1 and r 2 ). A part of the generator of the original cone is now the generator of the truncated cone. Denote it by the letter l.
The following relationship exists between the marked linear parameters:
l 2 = h 2 + (r 1 -r 2 ) 2 .
The side surface of a truncated figure
It was said above that it represents a lateral surface for a truncated cone. Cutting it along one of the generators, we obtain the following result.
Two circles represent the grounds. A quadrangular figure bounded by two straight segments and two arcs is the desired lateral surface of the truncated cone, the area of which must be found. We will solve this problem.
Note that this surface is a sector of a circle in which the central part is cut. Denote the radius of the outer arc by g. Then the radius of the inner arc will be g - l. Using the results of solving the previous proportion in determining the angle of the sector x, we can write the following equality:
x = 2 * pi * r 1 / g = 2 * pi * r 2 / (gl) =>
g = l * r 1 / (r 1 -r 2 ).
The desired area S b is equal to the difference in the areas of the sectors constructed using the radii g and gl. Using the formula for the sector area obtained above, you can write:
S b = pi * g * r 1 - pi * (gl) * r 2 .
Substituting the formula for g into this expression, we obtain finite equality for the area of the lateral surface of the truncated cone:
S b = pi * l * (r 1 + r 2 ).
The task of determining the area of the conical surface
We solve a simple problem. It is necessary to find the area of the lateral surface of the cone, if it is known that its height h is equal to the diameter of the base, and the generator is 15 cm.
We write the corresponding formula for S b , from which it will be seen which values should be calculated. We have:
S b = pi * r * g.
The value of the generator g is known from the conditions of the problem. It remains to determine the radius of the figure.
The generator, height and radius are related to each other by the following equality:
g 2 = h 2 + r 2 .
It follows from the condition that 2 * r = h. Substituting the value of h in the expression, we get:
g 2 = (2 * r) 2 + r 2 = 5 * r 2 =>
r = g / √5.
Now we substitute the formula for the radius of the base in the expression for S b , we get:
S b = pi / √5 * g 2 .
We have obtained the final formula, which shows that the area of the desired surface depends only on the length of the generatrix. Substituting g = 15 cm, we obtain the answer to the problem: S b ≈ 315.96 cm 2 .