Cylinder (circular cylinder) - a body that consists of two circles, superimposed parallel transfer, and all segments connecting the corresponding points of these circles. The circles are called the bases of the cylinder, and the line segments connecting the corresponding points of the circles of the circles are called the generatrices of the cylinder.
The bases of the cylinder are equal and lie in parallel planes, and the generatrices of the cylinder are parallel and equal. The surface of the cylinder consists of bases and a side surface. The lateral surface is formed by generators.
A cylinder is called straight if its generatrices are perpendicular to the base planes. A cylinder can be viewed as a solid obtained by rotating a rectangle around one of its sides as an axis. There are other types of cylinder - elliptical, hyperbolic, parabolic. A prism is also considered a type of cylinder.
Figure 2 shows an inclined cylinder. Circles with centers O and O 1 are its bases.
Cylinder radius - radius of its base. The height of the cylinder is the distance between the planes of the bases. The axis of the cylinder is a straight line passing through the centers of the bases. It is parallel to the generatrix. The section of a cylinder by a plane passing through the axis of the cylinder is called the axial section. The plane passing through the generatrix of a straight cylinder and perpendicular to the axial section drawn through this generatrix is called the tangent plane of the cylinder.
A plane perpendicular to the axis of the cylinder intersects its lateral surface in a circle equal to the circumference of the base.
A prism inscribed in a cylinder is a prism whose bases are equal polygons inscribed in the bases of the cylinder. Its lateral ribs are generatrices of the cylinder. A prism is called circumscribed about a cylinder if its bases are equal polygons circumscribed about the bases of the cylinder. The planes of its faces touch the lateral surface of the cylinder.
The area of the lateral surface of the cylinder can be calculated by multiplying the length of the generatrix by the perimeter of the section of the cylinder by the plane perpendicular to the generatrix.
The lateral surface area of a straight cylinder can be found by its sweep. The unfolded cylinder is a rectangle with height h and length P, which is equal to the perimeter of the base. Consequently, the area of the lateral surface of the cylinder is equal to the area of its sweep and is calculated by the formula:
In particular, for a straight circular cylinder:
P = 2πR, and S b = 2πRh.
The total surface area of a cylinder is equal to the sum of the areas of its lateral surface and its bases.
For a straight circular cylinder:
S p = 2πRh + 2πR 2 = 2πR (h + R)
There are two formulas for finding the volume of an inclined cylinder.
You can find the volume by multiplying the length of the generatrix by the cross-sectional area of the cylinder by the plane perpendicular to the generatrix.
The volume of the inclined cylinder is equal to the product of the base area by the height (the distance between the planes in which the bases lie):
V = Sh = S l sin α,
where l is the length of the generatrix, and α is the angle between the generatrix and the plane of the base. For a straight cylinder h = l.
The formula for finding the volume of a circular cylinder is as follows:
V = π R 2 h = π (d 2/4) h,
where d is the base diameter.
site, with full or partial copying of the material, a link to the source is required.
Cylinder(Old Greek. κύλινδρος - roller, roller) - a geometric body bounded by a cylindrical surface and two parallel planes intersecting it. A cylindrical surface is a surface obtained by such a translational motion of a straight line (generatrix) in space that the selected point of the generatrix moves along a plane curve (guideline). The part of the cylinder surface bounded by the cylindrical surface is called the lateral surface of the cylinder. The other part, limited by parallel planes, is the base of the cylinder. Thus, the border of the base will coincide in shape with the guide.
In most cases, a cylinder means a straight circular cylinder, in which the guide is a circle and the bases are perpendicular to the generatrix. Such a cylinder has an axis of symmetry.
Other types of cylinder - (by the inclination of the generatrix) oblique or inclined (if the generatrix does not touch the base at a right angle); (in the shape of the base) elliptic, hyperbolic, parabolic.
A prism is also a type of cylinder - with a polygon base.
Cylinder surface area
Side surface area
Calculating the lateral surface area of a cylinder
The area of the lateral surface of the cylinder is equal to the length of the generatrix multiplied by the perimeter of the section of the cylinder by the plane perpendicular to the generatrix.
The lateral surface area of a straight cylinder is calculated from its sweep. The unfolded cylinder is a rectangle with a height and length equal to the perimeter of the base. Consequently, the area of the lateral surface of the cylinder is equal to the area of its sweep and is calculated by the formula:
In particular, for a straight circular cylinder:
, andFor an inclined cylinder, the lateral surface area is equal to the length of the generatrix multiplied by the perimeter of the section perpendicular to the generatrix:
Unfortunately, there is no simple formula expressing the lateral surface area of an oblique cylinder through the parameters of the base and height, in contrast to the volume.
Total surface area
The total surface area of a cylinder is equal to the sum of the areas of its lateral surface and its bases.
For a straight circular cylinder:
Cylinder volume
There are two formulas for an inclined cylinder:
where is the length of the generatrix, and is the angle between the generatrix and the plane of the base. For straight cylinder.For a straight cylinder, and, and the volume is equal to:
For a circular cylinder:
where d- base diameter.
Notes (edit)
Wikimedia Foundation. 2010.
Synonyms:See what "Cylinder" is in other dictionaries:
- (lat. cylindrus) 1) a geometrical body, bounded from the ends by two circles, from the sides by a plane enveloping these circles. 2) in watchmaking: a special kind of double wheel lever. 3) a hat shaped like a cylinder. Dictionary of foreign words, ... ... Dictionary of foreign words of the Russian language
cylinder- a, m. cylindre m., German. Zylinder, lat. cylindrus c. 1. Geometric body formed by the rotation of a rectangle around one of its sides. Cylinder volume. BASS 1. The thickness of the cylinder is equal to the area of its base multiplied by the height. Dahl ... Historical Dictionary gallicisms of the Russian language
Husband., Greek. straight stack, shaft; blaze, blaze; a body bounded at the ends by two circles, and at the sides by a plane curved in circles. The thickness of the cylinder is equal to the area of its base multiplied by the height, geom. Steam cylinder, freebie, pipe in which ... ... Dahl's Explanatory Dictionary
Cylindrical surface, drum, shaft; Gibus, hat, roller, roll, mandrel, cylinder, point, tsarga, body, roller Dictionary of Russian synonyms. cylinder noun, number of synonyms: 22 attackostels (2) ... Synonym dictionary
- (from the Greek kylindros) in elementary geometry, a geometric body formed by rotating a rectangle about one side: the volume of the cylinder is V =? r2h, and the lateral surface area is S = 2? rh. The lateral surface of the cylinder is a part of the cylindrical ... ...
A hollow part with a cylindrical inner surface in which the piston moves. One of the main parts of piston machines and mechanisms ... Big Encyclopedic Dictionary
Tall men's hat made of silk plush with small firm brims ... Big Encyclopedic Dictionary
CYLINDER, solid or surface formed by rotating a rectangle around one of its sides as an axis. The volume of the cylinder, if we denote its height as h, and the radius of the base as r, is equal to pr2h, and the area of the curved surface is 2prh ... Scientific and technical encyclopedic dictionary
CYLINDER, cylinder, husband. (from the Greek kylindros). 1. A geometric body formed by the rotation of a rectangle about one of its sides, called an axis, and having a circle at its bases (mat.). 2. Part of machines (motors, pumps, compressors, etc.) in ... ... Ushakov's Explanatory Dictionary
CYLINDER, ah, husband. 1. A geometric body formed by rotating a rectangle around one of its sides. 2. Column-shaped object, eg. part of a piston machine. 3. Tall hard hat of this shape with small brim. Black c. | adj. ... ... Ozhegov's Explanatory Dictionary
- (Steam cylinder) is one of the main parts of piston machines. It is carried out in the form of a hollow round cylinder, in which the piston moves. The central heating of steam engines is usually supplied with a steam jacket to heat its walls in order to reduce steam condensation. ... ... Marine dictionary
Cylinder(more precisely, a circular cylinder) is a body that consists of two circles lying in parallel planes and combined by a parallel translation, and all segments connecting the corresponding points of these circles. The circles are called cylinder bases, and the segments connecting the corresponding points of the circles are generators.
The cylinder has the following properties, resulting from the fact that the bases of the cylinder are aligned by parallel translation:
1. The bases of the cylinder are equal.
2. The generatrices of the cylinder are parallel and equal.
The cylinder is called direct if its generatrices are perpendicular to the planes of the bases. In what follows, we will consider mainly straight cylinders, therefore, unless otherwise stated, by a cylinder we mean a straight cylinder.
Radius a cylinder is called the radius of its base. Height a cylinder is the distance between the planes of its bases. For a straight cylinder, the height is equal to the generatrix. Axis cylinder is called a straight line passing through the centers of the bases.
A cylinder is a body of revolution, since it can be obtained by rotating a rectangle around its axis.
Tasks
18.1 The height of the cylinder is 6, the radius of the base is 5. The ends of the line segment equal to 10 lie on the circumferences of both bases. Find the shortest distance from this segment to the axis of the cylinder.
18.2 In an equilateral cylinder (the diameter is equal to the height of the cylinder), the circumferential point of the upper base is connected to the circumferential point of the lower base. The angle between the radii drawn at these points is 60 °. Find the angle between the drawn line and the axis of the cylinder.
Cone
Defining a cone
Cone(more precisely, a circular cone) is a body that consists of a circle - base of the cone, a point not lying in the plane of the base, - cone tops and all line segments connecting the top of the cone to the base points. The segments connecting the vertices of the cone with the points of the base circle are called generators of the cone.
Cone height is called the perpendicular dropped from the top of the cone to the plane of the base. If the base of the height coincides with the center of the circle of the base, the cone is called direct... In what follows, a cone will usually be understood as a straight cone.
Axis a straight circular cone is called a straight line containing its height. Such a cone can be obtained by rotating right triangle around one of the legs.
Frustum
A plane parallel to the base of the cone cuts off a similar cone from it. The rest is called truncated cone.
Tasks
19.1Two generatrices of the cone, resting on the ends of the base diameter, make an angle of 60 ° between themselves. The radius of the cone is 3. Find the generatrix of the cone and its height.
19.2A straight line is drawn through the middle of the height of the cone, parallel to the generatrix. Find the length of a line segment inside a cone.
19.3 The generatrix of the cone is 13, height 12. The cone is crossed by a straight line parallel to the base; the distance from it to the base is 6, and to the height - 2. Find the line segment enclosed inside the cone.
19.4 The radii of the bases of the truncated cone are equal to 3 and 6, the height is 4. Find the generatrix.
Ball definition
Ball a body is called, which consists of all points in space located at a distance not more than a given from some point, called center of the ball... This distance is called ball radius.
The boundary of the ball is called ball surface or sphere... Thus, the points of the sphere are all points of the sphere that are removed from the center of the sphere by a distance equal to the radius.
The segment connecting two points of the ball surface and passing through the center of the ball is called the diameter of the ball.
A ball, like a cylinder and a cone, is a body of revolution. It is obtained by rotating a semicircle around its diameter.
Tasks
20.1 Three points are given on the surface of the ball. The straight-line distances between them are 6, 8 and 10. The radius of the ball is 13. Find the distance from the center of the ball to the plane passing through these three points.
20.2 The diameter of the sphere 25. On its surface a point and a circle are given, all points of which are removed (in a straight line) from by 15. Find the radius of this circle.
20.3 The radius of the ball is 7. On its surface, two circles are given that have a common chord of length 2. Find the radii of the circles, knowing that their planes are perpendicular.
A cylinder is a geometric body bounded by two parallel planes and a cylindrical surface. In this article, we will talk about how to find the area of a cylinder and, using the formula, we will solve several problems for example.
A cylinder has three surfaces: top, bottom, and flank.
The top and bottom of a cylinder are circles and are easy to identify.
It is known that the area of a circle is equal to πr 2. Therefore, the formula for the area of two circles (the top and bottom of the cylinder) will be πr 2 + πr 2 = 2πr 2.
The third, lateral surface of the cylinder, is the curved wall of the cylinder. In order to better represent this surface, let's try to transform it to get a recognizable shape. Imagine that the cylinder is an ordinary tin can that does not have a top lid and a bottom. Let's make a vertical cut on the side wall from the top to the bottom of the can (Step 1 in the picture) and try to open (straighten) the resulting figure as much as possible (Step 2).
After fully opening the resulting jar, we will see the already familiar shape (Step 3), this is a rectangle. The area of a rectangle is easy to calculate. But before that, let's go back for a moment to the original cylinder. The top of the original cylinder is a circle, and we know that the circumference is calculated by the formula: L = 2πr. It is marked in red in the figure.
When the side wall of the cylinder is fully opened, we see that the circumference becomes the length of the resulting rectangle. The sides of this rectangle will be the circumference (L = 2πr) and the height of the cylinder (h). The area of a rectangle is equal to the product of its sides - S = length x width = L x h = 2πr x h = 2πrh. As a result, we have obtained a formula for calculating the area of the lateral surface of a cylinder.
Formula of the lateral surface area of a cylinder
S side. = 2πrh
Cylinder full surface area
Finally, if we add up the areas of all three surfaces, we get the formula for the total surface area of a cylinder. The surface area of the cylinder is equal to the area of the top of the cylinder + the area of the base of the cylinder + the area of the lateral surface of the cylinder or S = πr 2 + πr 2 + 2πrh = 2πr 2 + 2πrh. Sometimes this expression is written with the identical formula 2πr (r + h).
The formula for the total surface area of a cylinder
S = 2πr 2 + 2πrh = 2πr (r + h)
r is the radius of the cylinder, h is the height of the cylinder
Examples of calculating the surface area of a cylinder
To understand the above formulas, let's try to calculate the surface area of a cylinder using examples.
1. The radius of the base of the cylinder is 2, the height is 3. Determine the area of the lateral surface of the cylinder.
The total surface area is calculated by the formula: S side. = 2πrh
S side. = 2 * 3.14 * 2 * 34.6. Total ratings received: 990.
A cylindrical surface is formed by moving a straight line parallel to itself. The point of the straight line that is highlighted moves along the specified planar curve - guide... This line is called generatrix of a cylindrical surface.
Straight cylinder is a cylinder in which the generatrices are perpendicular to the base. If the generatrices of the cylinder are not perpendicular to the base, then this will be tilt cylinder.
Circular cylinder- a cylinder whose base is a circle.
Round cylinder- a cylinder that is both straight and circular.
Straight circular cylinder determined by the radius of the base R and generating L which is equal to the height of the cylinder H.
A prism is a special case of a cylinder.
Formulas for finding the elements of a cylinder.
The lateral surface area of a straight circular cylinder:
S side = 2πRH
Total surface area of a straight circular cylinder:
S = Sside+ 2Smain = 2 π R (H + R)
The volume of a straight circular cylinder:
V = S main H = πR 2 H
A straight circular cylinder with a beveled base or a short beveled cylinder is defined by the radius of the base R, minimum height h 1 and maximum height h 2.
Beveled cylinder lateral surface area:
S side = πR (h 1 + h 2)
The area of the base of the beveled cylinder.
