The calculation of the areas of figures in space and on the plane is one of the priority problems of geometry, which has practical application. To calculate the area using formulas, the mathematical form of which is determined by the shape of the figure. In this article, we give formulas for the area of the lateral surface of a cone of ordinary and truncated.
Cone as a geometric figure
In the general case, a cone is a figure that is formed by a conical surface and base. You can draw it in the following way: suppose that there is some smooth plane curve (circle, parabola, and so on), if we connect all the points of this curve with some unique point in space, we can get a conical surface.
In most cases, in practice and when solving problems by stereometry, a specific type of cone is considered - a straight figure with a round base. By its name, it is clear that the curve on which the conical surface rests is a circle. If the perpendicular is lowered from the top of the conical surface to the base, then it will intersect it in the center of the circle. This is a sufficient sign to consider the cone straight. The described figure is shown in the figure below.
Round cone: lateral surface area
The formula for the surface area of the lateral figure under consideration can be obtained by understanding what is at stake. As mentioned above, any cone consists of two parts:
The question of finding the area of the base is reduced to calculating this value for the circle. For a conical surface, which is lateral for the figure, the solution of the problem is not trivial.
How to find the lateral surface area of a cone? The formula can be written if you expand the surface of the surface in question. This is done simply: the conical surface is cut along the generatrix (the segment connecting the top and the border of the base of the figure) and unfolds. The result of these actions is shown in the figure.
It can be seen that the conical surface turned into a circular flat sector. The figure shows the angle of this sector in degrees. It is determined by the ratio of the radius r of the circle located at the base and the generatrix a. Without going into mathematical calculations, we write the formula for the surface area of the side cone:
S = pi * r * a.
That is, the desired area depends on two parameters: the length of the radius and the generatrix.
The lateral surface of the truncated cone
Now we give the formula for the area of the lateral surface of a truncated cone. This figure is obtained if the upper part of the usual cone is cut off with a plane, while the cutting plane should be parallel to the circular base. The figure will be formed by two bases and one side surface.
To determine the surface area of the side, we perform the same operations to obtain a sweep of the figure as in the case of a conventional cone. We get the following picture.
The lateral surface is represented by a circular sector, in which the center is cut. The formula for its area is:
S = pi * g * (r + r ').
The corresponding values are indicated in the figure.
The task of determining the area of an ordinary cone
It is known that a round cone has a radius of 10 cm and a height of 12 cm. It is necessary to find the total surface area.
The required area has two components. For the foundation, this value will be:
S o = pi * r² = 3.14 * 10² = 314 cm².
Before calculating the lateral surface, calculate the length of the generatrix a of the figure. Considering the corresponding right-angled triangle, we can write the equality:
a = √ (r² + h²) = √ (10² + 12²) = 15.62 cm.
Now you can use the written formula for the conical surface area:
S b = pi * r * a = 3.14 * 10 * 15.62 ≈ 490.47 cm².
The full area of the figure is found by adding S o and S b . Performing the indicated action, we get the answer: S ≈ 804.5 cm².