Formulas for calculating the diameter of the cone. An example of solving a geometric problem

The shape of the cone is the object of study of stereometry. The main properties of the cone are its volume and surface area, which can be calculated using linear parameters. One of them is the diameter of the cone. In this article, we show how this diameter can be calculated from other known characteristics of the figure.

Round straight cone

In the general case, a cone is a figure constructed as a result of the movement of a segment along a certain curve in the plane, while the second end of the segment is fixed at a certain point in space. The segment itself is called the generatrix, or generatrix, and the curve is called the directrix, or guide.

According to the above definition, the curve that limits the shape can be of any kind at all. The most famous of them are parabola, hyperbola, ellipse and circle. In the latter case, they talk about a round cone.

The round cone can be inclined and straight. Both figures are shown in the figure below.

Here r is the radius of the circle that borders the base of the figure. The letter h denotes the height, which represents the perpendicular lowered to the base from the top of the cone. The letter a denotes the axis of the cone. It can be seen that in the case of a straight figure, its height coincides with the axis, that is, it intersects the circle in its center.

In addition to the radius r and height h, an important linear parameter of the cone is the length of its generator g. As was said, the generatrix is ​​a segment connecting the director with the height. For a straight round cone, all generators are equal to each other.

Further in the article, revealing the question of how to find the diameter of the cone, only the cone is round and straight.

Calculation of the diameter of the figure through linear parameters and the angle at the base

The described spatial figure can be obtained by rotating a right triangle around any leg. This fact is illustrated in the figure below.

The figure shows that the two legs AC and AB are the radius r and height h of the volume figure, respectively. Generatrix g is the hypotenuse of BC. These correspondences allow us to write the formula for the diameter of the cone through the known g and h:

d = 2 * √ (g ² - h ² )

When writing this formula, the Pythagorean theorem was used, as well as the determination of the diameter, which is twice as large as the radius of the base of the cone.

If the angle φ between the base and any of the figures forming g is known, then the diameter of the cone can be determined by the following formulas:

d = 2 * g * cos (φ);

d = 2 * h / tg (φ)

Both equalities are a consequence of the application of the definition of trigonometric functions of the tangent and cosine.

Calculation of diameter through surface area and generatrix

The surface of the cone in question is formed by a conical surface and a circular base. A cone scan is shown below.

The total sweep area is determined by the following formula:

S = pi * r ² + pi * r * g

If the area S and the generator g are known, then this equation allows you to calculate the radius of the figure, and hence its diameter. Note that this is a second-order equation with respect to radius r. It should be solved using the discriminant. When solving, as a rule, two roots are obtained, one of which is negative. It must be discarded because of its non-physical significance.

Using the described technique, the problem will be solved at the end of the article, and an answer will be received to the question of what the diameter of the cone is.

Determination of diameter through volume and height

Now we show how to find the diameter of the cone, knowing its volume V and height h. To do this, remember that the volume of the cone, like the volume of any pyramid, can be determined using the following equality:

V = 1/3 * S * h

Here S is the area of ​​the base. Since the base area in this case is the area of ​​a circle, this expression can be rewritten in the following form:

V = 1/3 * pi * r ² * h

It remains to express the radius from here and double it, and we will get the answer to the question of how to find the diameter of the cone in terms of V and h. We have:

r = √ (3 * V / (pi * h));

d = 2 * r = 2 * √ (3 * V / (pi * h))

Note that the dimension of the length is obtained on the right-hand side. This proves the correctness of the obtained formula.

All formulas written in the article for the diameter d of the figure are also valid for a radius that will be half the diameter.

The task of determining the diameter through the known area of ​​the cone and its generatrix

Dan cone, the surface area of which is 150 cm ² . Generatrix is ​​equal to 14 cm. What is the diameter of the cone?

To get an answer to this question, we use the technique described in the article. First, we write the corresponding equation:

S = pi * r ² + pi * r * g =>

r ² + 14 * r - 150 / 3,14 = 0

Upon receipt of the last equality, we divided its left and right parts by the number Pi. We calculate the discriminant D. We have:

D = 14 ² - 4 * 1 * (- 150 / 3,14) = 387,0828

The resulting discriminant is accurate to 0.0001. The formula for the roots of the equation r has the following form:

r = (-14 ± √D) / 2

Obviously, one of the roots will be negative. We will not calculate it. We define only the desired positive radius of the figure:

r = (-14 + √387.0828) / 2 = 2.837 cm

To find the diameter of the cone, it remains to multiply this value by two and write down the answer: d = 5.674 cm.

In the end, we note that, knowing any two parameters of the round cone of a straight line, you can determine any of its characteristics, including volume and surface area.