A cylinder (derived from the Greek language, from the words "skating rink", "roller") is a geometric body that is bounded on the outside by a surface called a cylindrical surface and two planes. These planes intersect the surface of the figure and are parallel to each other.

A cylindrical surface is a surface that is obtained by a straight line in space. These movements are such that the selected point of this straight line moves along a flat-type curve. Such a straight line is called a generatrix, and a curved line is called a guide.

The cylinder consists of a pair of bases and a lateral cylindrical surface. Cylinders are of several types:

1. Circular, straight cylinder. For such a cylinder, the base and the guide are perpendicular to the generatrix, and there is

2. Inclined cylinder. He has an angle between the generating line and the base is not straight.

3. A cylinder of a different shape. Hyperbolic, elliptical, parabolic and others.

The area of ​​a cylinder, as well as the total surface area of ​​any cylinder, is found by adding the areas of the bases of this figure and the area of ​​\u200b\u200bthe lateral surface.

The formula for calculating the total area of ​​a cylinder for a circular, straight cylinder is:

Sp = 2p Rh + 2p R2 = 2p R (h+R).

The area of ​​the lateral surface is a little more difficult to find than the area of ​​the entire cylinder; it is calculated by multiplying the length of the generatrix by the perimeter of the section formed by the plane that is perpendicular to the generatrix.

The cylinder data for a circular, straight cylinder is recognized by the development of this object.

A development is a rectangle that has height h and length P, which is equal to the perimeter of the base.

It follows that the lateral area of ​​the cylinder is equal to the area of ​​the sweep and can be calculated using this formula:

If we take a circular, straight cylinder, then for it:

P = 2p R, and Sb = 2p Rh.

If the cylinder is inclined, then the lateral surface area must be equal to the product of the length of its generatrix and the perimeter of the section, which is perpendicular to this generatrix.

Unfortunately, there is no simple formula for expressing the lateral surface area of ​​an inclined cylinder in terms of its height and its base parameters.

To calculate a cylinder, you need to know a few facts. If a section with its plane intersects the bases, then such a section is always a rectangle. But these rectangles will be different, depending on the position of the section. One of the sides of the axial section of the figure, which is perpendicular to the bases, is equal to the height, and the other is equal to the diameter of the base of the cylinder. And the area of ​​such a section, respectively, is equal to the product of one side of the rectangle by the other, perpendicular to the first, or the product of the height of this figure by the diameter of its base.

If the section is perpendicular to the bases of the figure, but does not pass through the axis of rotation, then the area of ​​\u200b\u200bthis section will be equal to the product of the height of this cylinder and a certain chord. To get a chord, you need to build a circle at the base of the cylinder, draw a radius and set aside on it the distance at which the section is located. And from this point you need to draw perpendiculars to the radius from the intersection with the circle. The intersection points are connected to the center. And the base of the triangle is the desired one, which is searched for sounds like this: “The sum of the squares of two legs is equal to the hypotenuse squared”:

C2 = A2 + B2.

If the section does not affect the base of the cylinder, and the cylinder itself is circular and straight, then the area of ​​\u200b\u200bthis section is found as the area of ​​​​the circle.

The area of ​​a circle is:

S env. = 2p R2.

To find R, you need to divide its length C by 2p:

R = C \ 2n, where n is pi, a mathematical constant calculated to work with circle data and equal to 3.14.

A cylinder is a figure consisting of a cylindrical surface and two circles arranged in parallel. Calculating the area of ​​a cylinder is a problem in the geometric branch of mathematics, which is solved quite simply. There are several methods for solving it, which as a result always come down to one formula.

How to find the area of ​​a cylinder - calculation rules

• To find out the area of ​​\u200b\u200bthe cylinder, it is necessary to add two base areas to the area of ​​\u200b\u200bthe lateral surface: S \u003d S side. In a more detailed version, this formula looks like this: S= 2 π rh+ 2 π r2= 2 π r(h+ r).
• The lateral surface area of ​​a given geometric body can be calculated if its height and the radius of the circle underlying the base are known. In this case, you can express the radius from the circumference, if it is given. The height can be found if the value of the generatrix is ​​specified in the condition. In this case, the generatrix will be equal to the height. The formula for the lateral surface of a given body looks like this: S= 2 π rh.
• The area of ​​the base is calculated according to the formula for finding the area of ​​a circle: S osn= π r 2 . In some problems, the radius may not be given, but the circumference is given. With this formula, the radius is expressed quite easily. С=2π r, r= С/2π. It must also be remembered that the radius is half the diameter.
• When performing all these calculations, the number π is usually not translated into 3.14159 ... You just need to add it next to the numerical value that was obtained as a result of the calculations.
• Further, it is only necessary to multiply the found area of ​​\u200b\u200bthe base by 2 and add to the resulting number the calculated area of ​​\u200b\u200bthe lateral surface of the figure.
• If the problem indicates that the cylinder has an axial section and this is a rectangle, then the solution will be slightly different. In this case, the width of the rectangle will be the diameter of the circle that lies at the base of the body. The length of the figure will be equal to the generatrix or the height of the cylinder. It is necessary to calculate the desired values ​​​​and substitute in an already known formula. In this case, the width of the rectangle must be divided by two to find the area of ​​the base. To find the side surface, the length is multiplied by two radii and by the number π.
• You can calculate the area of ​​a given geometric body through its volume. To do this, you need to derive the missing value from the formula V=π r 2 h.
• There is nothing difficult in calculating the area of ​​a cylinder. You only need to know the formulas and be able to derive from them the quantities necessary for the calculations.

A cylinder is a geometric body bounded by two parallel planes and a cylindrical surface. In the 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: a top, a bottom, and a side surface.

The top and bottom of the cylinder are circles and are easy to define.

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 look like πr 2 + πr 2 = 2πr 2 .

The third, side 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 a cylinder is an ordinary tin can that does not have a top lid and bottom. Let's make a vertical incision on the side wall from the top to the bottom of the jar (Step 1 in the figure) and try to open (straighten) the resulting figure as much as possible (Step 2).

After the full disclosure of the resulting jar, we will see a familiar figure (Step 3), this is a rectangle. The area of ​​a rectangle is easy to calculate. But before that, let us return for a moment to the original cylinder. The vertex of the original cylinder is a circle, and we know that the circumference of a circle is calculated by the formula: L = 2πr. It is marked in red in the figure.

When the side wall of the cylinder is fully expanded, 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 lateral surface area of ​​a cylinder.

The formula for the area of ​​the lateral surface of a cylinder
S side = 2prh

Full surface area of ​​a cylinder

Finally, if we add up the area 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 side surface of the cylinder or S = πr 2 + πr 2 + 2πrh = 2πr 2 + 2πrh. Sometimes this expression is written by 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 side surface of the cylinder.

The total surface area is calculated by the formula: S side. = 2prh

S side = 2 * 3.14 * 2 * 3

S side = 6.28 * 6

S side = 37.68

The lateral surface area of ​​the cylinder is 37.68.

2. How to find the surface area of ​​a cylinder if the height is 4 and the radius is 6?

The total surface area is calculated by the formula: S = 2πr 2 + 2πrh

S = 2 * 3.14 * 6 2 + 2 * 3.14 * 6 * 4

S = 2 * 3.14 * 36 + 2 * 3.14 * 24

It is a geometric body bounded by two parallel planes and a cylindrical surface.

The cylinder consists of a side surface and two bases. The formula for the surface area of ​​a cylinder includes a separate calculation of the area of ​​the bases and the lateral surface. Since the bases in the cylinder are equal, then its total area will be calculated by the formula:

We will consider an example of calculating the area of ​​\u200b\u200ba cylinder after we know all the necessary formulas. First we need the formula for the area of ​​the base of a cylinder. Since the base of the cylinder is a circle, we need to apply:
We remember that these calculations use a constant number Π = 3.1415926, which is calculated as the ratio of the circumference of a circle to its diameter. This number is a mathematical constant. We will also consider an example of calculating the area of ​​​​the base of a cylinder a little later.

Cylinder side surface area

The formula for the area of ​​the lateral surface of a cylinder is the product of the length of the base and its height:

Now consider a problem in which we need to calculate the total area of ​​a cylinder. In a given figure, the height is h = 4 cm, r = 2 cm. Let's find the total area of ​​the cylinder.
First, let's calculate the area of ​​the bases:
Now consider an example of calculating the lateral surface area of ​​a cylinder. When expanded, it is a rectangle. Its area is calculated using the above formula. Substitute all the data into it:
The total area of ​​a circle is the sum of twice the area of ​​the base and the side:

Thus, using the formulas for the area of ​​the bases and the lateral surface of the figure, we were able to find the total surface area of ​​the cylinder.
The axial section of the cylinder is a rectangle in which the sides are equal to the height and diameter of the cylinder.

The formula for the area of ​​the axial section of a cylinder is derived from the calculation formula:

Find the area of ​​the axial section perpendicular to the bases of the cylinder. One of the sides of this rectangle is equal to the height of the cylinder, the other is equal to the diameter of the base circle. Accordingly, the cross-sectional area in this case will be equal to the product of the sides of the rectangle. S=2R*h, where S is the cross-sectional area, R is the radius of the base circle, given by the conditions of the problem, and h is the height of the cylinder, also given by the conditions of the problem.

If the section is perpendicular to the bases, but does not pass through the axis of rotation, the rectangle will not equal the diameter of the circle. It needs to be calculated. To do this, the task must say at what distance from the axis of rotation the section plane passes. For the convenience of calculations, construct a circle of the base of the cylinder, draw a radius and set aside on it the distance at which the section is located from the center of the circle. From this point, draw to the perpendiculars until they intersect with the circle. Connect the intersection points to the center. You need to find chords. Find the size of half a chord using the Pythagorean theorem. It will be equal to the square root of the difference of the squares of the radius of the circle from the center to the section line. a2=R2-b2. The whole chord will be, respectively, equal to 2a. Calculate the cross-sectional area, which is equal to the product of the sides of the rectangle, that is, S=2a*h.

The cylinder can be dissected without passing through the plane of the base. If the cross section is perpendicular to the axis of rotation, then it will be a circle. Its area in this case is equal to the area of ​​​​the bases, that is, it is calculated by the formula S \u003d πR2.

Useful advice

To more accurately imagine the section, make a drawing and additional constructions to it.

Sources:

• cylinder cross section area

The line of intersection of a surface with a plane belongs both to the surface and to the secant plane. The line of intersection of a cylindrical surface with a secant plane parallel to the straight generatrix is ​​a straight line. If the cutting plane is perpendicular to the axis of the surface of revolution, the section will have a circle. In general, the line of intersection of a cylindrical surface with a cutting plane is a curved line.

You will need

• Pencil, ruler, triangle, patterns, compasses, measuring instrument.

Instruction

On the frontal projection plane P₂, the section line coincides with the projection of the secant plane Σ₂ in the form of a straight line.
Designate the points of intersection of the generators of the cylinder with the projection Σ₂ 1₂, 2₂, etc. to points 10₂ and 11₂.

On the plane P₁ is a circle. Points 1₂ , 2₂ marked on the section plane Σ₂, etc. with the help of a projection line, the connections will be projected on the outline of this circle. Designate their horizontal projections symmetrically about the horizontal axis of the circle.

Thus, the projections of the desired section are defined: on the plane P₂ - a straight line (points 1₂, 2₂ ... 10₂); on the plane P₁ - a circle (points 1₁, 2₁ ... 10₁).

By two, construct the natural size of the section of the given cylinder by the front-projecting plane Σ. To do this, use the method of projections.

Draw the plane P₄ parallel to the projection of the plane Σ₂. On this new x₂₄ axis, mark the point 1₀. Distances between points 1₂ - 2₂, 2₂ - 4₂, etc. from the frontal projection of the section, set aside on the x₂₄ axis, draw thin lines of projection connection perpendicular to the x₂₄ axis.

In this method, the P₄ plane is replaced by the P₁ plane, therefore, from the horizontal projection, transfer the dimensions from the axis to the points to the axis of the P₄ plane.

For example, on P₁ for points 2 and 3, this will be the distance from 2₁ and 3₁ to the axis (point A), etc.

Having postponed the indicated distances from the horizontal projection, you will get points 2₀, 3₀, 6₀, 7₀, 10₀, 11₀. Then, for greater accuracy of construction, the remaining, intermediate, points are determined.

By connecting all the points with a curved curve, you will obtain the desired natural size of the cross section of the cylinder by the front-projecting plane.

Sources:

• how to replace plane

Tip 3: How to find the area of ​​the axial section of a truncated cone

To solve this problem, you need to remember what a truncated cone is and what properties it has. Be sure to draw. This will determine which geometric figure is a section. It is quite possible that after this the solution of the problem will no longer be difficult for you.

Instruction

A round cone is a body obtained by rotating a triangle around one of its legs. Straight lines coming from the top cones and intersecting its base are called generators. If all generators are equal, then the cone is straight. At the base of the round cones lies a circle. The perpendicular dropped to the base from the top is the height cones. At the round straight cones height coincides with its axis. The axis is a straight line connecting to the center of the base. If the horizontal cutting plane of the circular cones, then its upper base is a circle.

Since it is not specified in the condition of the problem, it is the cone that is given in this case, we can conclude that this is a straight truncated cone, the horizontal section of which is parallel to the base. Its axial section, i.e. vertical plane, which through the axis of a circular cones, is an isosceles trapezoid. All axial sections round straight cones are equal to each other. Therefore, to find square axial sections, it is required to find square trapezoid, the bases of which are the diameters of the bases of the truncated cones, and the sides are its generators. Truncated Height cones is also the height of the trapezoid.

The area of ​​a trapezoid is determined by the formula: S = ½(a+b) h, where S is square trapezoid; a - the value of the lower base of the trapezoid; b - the value of its upper base; h - the height of the trapezoid.

Since the condition does not specify which ones are given, it is possible that the diameters of both bases of the truncated cones known: AD = d1 is the diameter of the lower base of the truncated cones;BC = d2 is the diameter of its upper base; EH = h1 - height cones.In this way, square axial sections truncated cones defined: S1 = ½ (d1+d2) h1

Sources:

• truncated cone area

The cylinder is a three-dimensional figure and consists of two equal bases, which are circles, and a lateral surface connecting lines bounding the bases. To calculate square cylinder, find the areas of all its surfaces and add them up.