The area of a truncated cone. Formula and problem example

2024 Author: Angel Austin | [email protected]. Last modified: 2023-12-17 05:14

The figures of revolution in geometry are given special attention when studying their characteristics and properties. One of them is a truncated cone. This article aims to answer the question of what formula can be used to calculate the area of a truncated cone.

Which figure are we talking about?

Before describing the area of a truncated cone, it is necessary to give an exact geometric definition of this figure. Truncated is such a cone, which is obtained as a result of cutting off the vertex of an ordinary cone by a plane. In this definition, a number of nuances should be emphasized. First, the section plane must be parallel to the plane of the base of the cone. Secondly, the original figure must be a circular cone. Of course, it can be an elliptical, hyperbolic and other type of figure, but in this article we will restrict ourselves to considering only a circular cone. The latter is shown in the figure below.

It's easy to guess that it can be obtained not only with the help of a section by a plane, but also with the help of a rotation operation. ForTo do this, you need to take a trapezoid that has two right angles and rotate it around the side that is adjacent to these right angles. As a result, the bases of the trapezoid will become the radii of the bases of the truncated cone, and the lateral inclined side of the trapezoid will describe the conical surface.

Shape development

Considering the surface area of a truncated cone, it is useful to bring its development, that is, the image of the surface of a three-dimensional figure on a plane. Below is a scan of the studied figure with arbitrary parameters.

It can be seen that the area of the figure is formed by three components: two circles and one truncated circular segment. Obviously, to determine the required area, it is necessary to add up the areas of all the named figures. Let's solve this problem in the next paragraph.

Truncated cone area

To make it easier to understand the following reasoning, we introduce the following notation:

• r1, r2 - radii of the large and small bases respectively;
• h - figure height;
• g - generatrix of the cone (the length of the oblique side of the trapezoid).

The area of the bases of a truncated cone is easy to calculate. Let's write the corresponding expressions:

S_o1=pir₁²;

S_o2=pir₂².

The area of a part of a circular segment is somewhat more difficult to determine. If we imagine that the center of this circular sector is not cut out, then its radius will be equal to the value G. It is not difficult to calculate it if we consider the correspondingsimilar right-angled cone triangles. It is equal to:

G=r₁g/(r₁-r₂).

Then the area of the whole circular sector, which is built on radius G and which relies on an arc of length 2pir₁, will be equal to:

S₁=pir₁G=pir₁ ²g/(r₁-r₂).

Now let's determine the area of the small circular sector S₂, which will need to be subtracted from S₁. It is equal to:

S₂=pir₂(G - g)=pir_{2 (r}1_g/(r1_-r2_{) - g)=pir}22^g/(r1_-r2 _).

The area of the conical truncated surface S_bis equal to the difference between S₁ and S₂. We get:

S_b=S₁- S₂=pir ₁²g/(r₁-r₂) - pi r₂²g/(r₁-r_2)=pig(r1_+r2_).

Despite some cumbersome calculations, we got a fairly simple expression for the area of the side surface of the figure.

Adding the areas of the bases and S_b, we arrive at the formula for the area of a truncated cone:

S=S_o1+ S_o2+ S_b=pir 12 ^{+ pir}22 ^{+ pig (r}1_+r2_).

Thus, to calculate the value of S of the studied figure, you need to know its three linear parameters.

Example problem

Circular straight conewith a radius of 10 cm and a height of 15 cm was cut off by a plane so that a regular truncated cone was obtained. Knowing that the distance between the bases of the truncated figure is 10 cm, it is necessary to find its surface area.

To use the formula for the area of a truncated cone, you need to find three of its parameters. One we know:

r₁=10 cm.

The other two are easy to calculate if we consider similar right-angled triangles, which are obtained as a result of the axial section of the cone. Taking into account the condition of the problem, we get:

r₂=105/15=3.33 cm.

Finally, the guide of the truncated cone g will be:

g=√(10²+ (r₁-r₂) ²)=12.02 cm.

Now you can substitute the values r₁, r₂ and g into the formula for S:

S=pir₁²+ pir₂ 2^{+ pig(r}1_+r2_{)=851.93 cm} 2.

The desired surface area of the figure is approximately 852 cm².