Concept
The principles of formation of a geometric body are simple. Let's imagine two parallel planes a and a1. From the one located at the first point, the perpendicular descends to the second. The point on a1 is the base of the perpendicular, it is the center of the circle. If you connect a point on the plane a with each point on the circle on a1, you get a cone. The base of its perpendicular is the height.
The second option for the formation of the geometric body under consideration: a right triangle rotates around a leg clockwise or counterclockwise. The leg, which became the axis, will be the height of the cone, the one lying at the base will be the diameter of the lower surface, and the hypotenuse will be the generatrix.
The length of the generatrices is the same; their totality is called the lateral surface. The square of the length of the generatrix is equal to the sum of the squares of the height and radius of the base (from the Pythagorean theorem): l2 = h2 + r2. From here
Elements of a figure and its linear characteristics
A truncated cone is a spatial figure consisting of three surfaces. Two of them are round bases (upper and lower) and one is a side surface. Unlike polyhedra, the figure in question does not have vertices or faces.
Important parameters of a truncated cone are the radii of each of the bases. We will denote the larger radius as r1, and the smaller radius as r2. In addition to the radii of the figure, to unambiguously determine it, it is necessary to know either the height h or the generator g. These parameters are related mathematically by the following equality:
g2 = h2 + (r1 - r2)2
All four parameters are used to determine surface area and volume.
Types of cones
In geometry, there are almost a dozen types of cones:
- Direct circular - the bottom edge is represented by a circle - a figure with a center of symmetry. The axis that runs from the top to the center of the base is perpendicular to the plane of the latter.
- Oblique or oblique - the projection of the vertex onto the lower surface does not coincide with its center.
- Circular - with a circle in the center.
- Straight - the lower surface is represented by a circle or ellipse. The center of the lower surface coincides with the projection of the vertex onto it.
- Hyperbolic, parabolic, elliptic - based on the corresponding figures.
- Equilateral – the generatrix is equal to the diameter of the lower surface.
- Truncated - limited to a plane parallel to the base. It is located between it and the top of the geometric body.
- Double - two identical bodies have a common top or base and axis - passes through both bodies.
Problem 3
Rice. 3. – desired angle
The area of the lateral surface of the cone is twice the area of its base. Find the angle between the generatrix of the cone and the plane of its base. Give your answer in degrees (see Fig. 3).
Solution
Means, .
Now let's look at the axial section and draw the height (axis). We obtain a right triangle in which the leg (radius of the base) is half the size of the hypotenuse, which means that the angle at the radius is 60 degrees (see Fig. 4).
Rice. 4. Illustration for the problem
Answer: 60 degrees.
Square
By area we mean the number of squares with a side of one that fit on a certain surface.
The area of a straight cone is determined by the formula:
- bases – Sbas = πr2; r – radius;
- lateral surface – Sbp = πrl; l – length;
- complete – S = Smain + Sbp = πr2 + πrl = πr (r + l).
Example:
- diameter is 12 cm;
- length of the generatrix – 10 cm.
Solution.
The radius is half the diameter: 12/2 = 6 cm.
Let's substitute the values into the expression: S = πr (r + l).
We get: S = π * 6 *(6 + 10) = 96 π ≈ 301.584 cm2.
How to construct a development of the surface of a right truncated cone
We divide the base of the cone into 12 equal parts (we fit in the correct pyramid). These construction elements are already ready from the drawing “Section of a cone by a plane of particular position.”
We construct a development of the lateral surface of the cone, which is a circular sector. The center of its radius is taken to be the vertex of the cone, and the radius of the circular sector of the cone is equal to the length of the generatrix of the cone, and the length of the arc of the sector is equal to the circumference of the base of the cone. We transfer 12 chords to the arc of the sector, which will determine its length, as well as the angle of the circular sector.
We attach the base of the cone to the central point of the arc of the lateral scan sector of the truncated cone. Its base is projected in full size onto the horizontal projection plane.
On the development of the cone, we attach the natural size of the section to its base.
The two outermost generatrices of the cone, which form its main contour, are projected onto the frontal projection plane in full size, so they can be immediately transferred to the development of the lateral surface of the cone. Since part of it is cut off by the frontally projecting plane, we transfer only the rightmost truncated generatrix to the development of the cone. The remaining truncated generatrices of the cone are projected onto the frontal plane of projections with distortion. Their natural size is found by rotating around the axis of the cone to a position parallel to the frontal plane of projections.
The very principle of finding the natural values of the generators of a truncated cone comes down to drawing a horizontal straight line from the points of intersection of the generators with the plane to the extreme right (left) generatrix and measuring their natural values on it. All actions are carried out on the frontal plane of projection.
On each generatrix lying on the development of the lateral surface of the cone, we plot the actual lengths of the truncated generators. We connect the resulting points with a smooth curved line using the Spline command in AutoCAD.
We completed the descriptive geometry task of constructing the development of a truncated cone, but in order to avoid problems during its defense (when I was studying, every coursework in descriptive geometry was defended), we will once again consider the principle of rotation for finding the natural size of the truncated generatrix of the cone.
“Their natural size is found by rotating around the axis of the cone to a position parallel to the frontal plane of projection.” When we rotate the generatrix of a straight cone to a position parallel to the frontal plane of projection, then its trajectory describes an arc on the horizontal plane of projection, and on the frontal straight line!
You don’t have to draw connection lines from the horizontal plane of projection to the frontal one, because it is obvious that the point will lie on the extreme main generatrix of the cone contour for each generatrix when finding its natural size. Therefore, the very principle of rotation to find the natural size of the generatrices of a cone comes down to drawing a horizontal straight line from the points of the truncated generatrices to the main generatrix of the contour of the cone.
The video tutorial shows very clearly and in detail the principle of constructing the development of a straight truncated cone.
Volume of a truncated cone
Volume is the space occupied by a geometric body. The numerical value indicates the number of cubes with a unit edge that fit in the cone. The volume of a body is calculated as one third of the product of the area of the base and its height.
The base is a circle, its surface is calculated by the formula: Sbas = πr2. After substitution we get:
.
Example: calculate the volume of a body: r = 6 cm, h = 9 cm. We put the values in the formula and simplify the expression step by step.
If the diameter is known, divide it by two: .
Let's calculate the volume of a truncated cone. To understand, from the total volume of the original body you need to subtract the value cut off by a plane parallel to the bottom face.
Formula for the volume of a truncated cone:
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Specialists of the Shkolkovo educational project offer a new approach to preparing for the exam, involving a transition from simple to complex. First, we give the full theory, basic formulas and elementary practical problems with solutions, including those on the topic “Cone,” and then gradually move on to expert-level tasks, which are also found in the Unified State Exam. All necessary information is presented in the “Theoretical Information” section.
You can also immediately start solving online problems for calculating the height of a truncated cone, its lateral surface area, volume, as well as similar calculation problems, for example, finding the volume or cross-sectional area of a cube. A large database of exercises is presented in the “Catalogue” section. The list of tasks is systematically updated.
Check how easily you can determine the area of a cone online. If the exercise required minimal effort from you, we recommend that you do not waste time on simple tasks and move on to more complex ones. And if difficulties still arise, then you should definitely find time in your daily schedule for distance learning with Shkolkovo. With us you can quickly learn an algorithm for solving problems involving calculating the volume of a cone and other unknown parameters.
Problem 4
Rice. 5. Cone section
Rice. 6. Figure at the base of the cone
The radius of the base of the cone is 6, and its height is 8. The section plane contains the vertex of the cone and the chord of the base, the length of which is 4 (see Fig. 5). Find the distance from the center of the base of the cone to the section plane.
Solution
Consider the base of the cone. Since the chord in the section plane is equal to 4, and the radius is equal to 6, we have an isosceles triangle, the height of which, according to the Pythagorean theorem, is equal (see Fig. 6).
Rice. 7. Final drawing
Consider a triangle ( – the vertex of the cone). Let us prove that the height of this triangle is the required distance. Firstly, by construction. Secondly, it is a plane (since it is perpendicular to and ), which means .
Therefore, is the required distance. According to the Pythagorean theorem. And then...
Answer: .
