Each student heard a cone about the figure. Its properties and characteristics are studied by stereometry. From this figure you can get its truncated version. In this article, we consider the question of what is a scan of a truncated cone and how to find its area.
Which figure will we study?
A round straight truncated cone is two circles having different diameters, which are located in parallel planes. The circles of these circles are connected by straight segments of equal length, referred to as forming figures. The distance between the round bases is called height. The described figure is shown below in the photo.
You can get it in two fundamentally different geometric ways. Firstly, you can take a regular round cone and cut off the upper part with a plane parallel to its base. This action will lead to the formation of the upper (small) base of the truncated cone. Secondly, you can take a trapezoid with two right angles and rotate it around the side bounded by these angles. The side of the trapezoid around which the rotation will take place is called the axis of the figure. Two parallel sides of the trapezoid will describe the round bases during rotation, and the fourth inclined side forms the side surface of the figure.
The diagram above demonstrates the preparation of a truncated cone using a plane section.
Truncated Cone Sweep
As we saw, the figure in question is formed by three surfaces. Two of them represent the base, and the third is lateral. The sum of the areas of these surfaces is the total surface of the truncated cone. In three-dimensional space, its area is inconvenient to calculate, since the quantity itself is two-dimensional. In this regard, if there is a problem of determining the surface area of spatial figures, they are usually represented on the plane.
In our case, it’s easy to get a scan. To do this, mentally cut off the corresponding circles of the base from the figure. Then, it is necessary to cut along the generatrix and reveal the lateral surface. The result is the result shown in the photo.
It represents two different circles and part of a circular sector, in which the center is cut out.
Formula surface area formula
To calculate the surface area of a truncated cone, it is necessary to determine this value for each part of its sweep. We denote the radii of the bases by the letters R and r. Then their areas will be equal:
S o1 = pi * R 2 ;
S o2 = pi * r 2 .
To calculate the area of the lateral surface, we take into account that its development is formed by two identical generatrices g and two arcs of circles that are 2 * pi * r and 2 * pi * R in length. Omitting the reasoning and intermediate mathematical formulas, we give the final expression for the area of this part of the sweep of the figure. It has the form:
S b = pi * g * (R + r).
Having obtained the areas for the bases and the lateral surface, we can write down the development formula for the truncated cone. Its total area S is equal to:
S = S o1 + S o2 + S b = pi * (R 2 + r 2 + g * (R + r)).
Thus, the area S of the figure is uniquely determined from the knowledge of the radii of its bases and the length of the generatrix.
Geometric problem
It is necessary to calculate the development of a truncated cone, which has a height of 13 cm and the radii of the bases 2 cm and 7 cm.
Solving this problem by directly applying the formula for S is impossible, since the length of the generator g is not known. However, it can be calculated using the following formula:
g = √ (h 2 + (Rr) 2 ).
This expression can be obtained independently by considering a right triangle with sides g, h and (Rr), here h is the height of the truncated cone. Generatrix g will be equal to 13.93 cm (the value is given with an accuracy of 0.01 cm).
It remains to substitute the values of the generatrix and the radii in the formula for S to get the desired answer:
S = 3.14 * (7 2 + 2 2 + 13.93 * (7 + 2)) ≈ 560.1 cm 2 .
It should not be forgotten that the formula written for S is valid only for a round straight truncated cone.