How to calculate the cross-sectional area of a cylindrical part. Tutorial: Cylinder
The name of the science “geometry” is translated as “earth measurement”. It originated through the efforts of the very first ancient land managers. And it happened like this: during the floods of the sacred Nile, streams of water sometimes washed away the boundaries of farmers’ plots, and the new boundaries might not coincide with the old ones. Taxes were paid by peasants to the treasury of the pharaoh in proportion to the size of the land allotment. Special people were involved in measuring the areas of arable land within the new boundaries after the spill. It was as a result of their activities that the new science, which was developed in Ancient Greece. There it received its name and practically acquired modern look. Subsequently, the term became an international name for the science of flat and three-dimensional figures.
Planimetry is a branch of geometry dealing with the study flat figures. Another branch of science is stereometry, which examines the properties of spatial (volumetric) figures. Such figures include the one described in this article - a cylinder.
Examples of the presence of cylindrical objects in Everyday life plenty. Almost all rotating parts - shafts, bushings, journals, axles, etc. - have a cylindrical (much less often - conical) shape. The cylinder is also widely used in construction: towers, support, decorative columns. And also dishes, some types of packaging, pipes of various diameters. And finally - the famous hats, which have long become a symbol of male elegance. The list goes on and on.
Definition of a cylinder as a geometric figure
A cylinder (circular cylinder) is usually called a figure consisting of two circles, which, if desired, are combined using parallel translation. These circles are the bases of the cylinder. But the lines (straight segments) connecting the corresponding points are called “generators”.
It is important that the bases of the cylinder are always equal (if this condition is not met, then we have - frustum, anything else, but not a cylinder) and are in parallel planes. The segments connecting corresponding points on circles are parallel and equal.
The set of an infinite number of constituents is nothing more than side surface cylinder - one of the elements of this geometric figure. Its other important component is the circles discussed above. They are called bases.
Types of cylinders
The simplest and most common type of cylinder is circular. It is formed by two regular circles acting as bases. But instead of them there may be other figures.
The bases of the cylinders can form (in addition to circles) ellipses and other closed figures. But the cylinder may not necessarily have a closed shape. For example, the base of a cylinder can be a parabola, a hyperbola, or another open function. Such a cylinder will be open or deployed.
According to the angle of inclination of the cylinders forming the bases, they can be straight or inclined. For a straight cylinder, the generatrices are strictly perpendicular to the plane of the base. If this angle is different from 90°, the cylinder is inclined.
What is a surface of revolution
The straight circular cylinder is without a doubt the most common surface of rotation used in engineering. Sometimes, for technical reasons, conical, spherical, and some other types of surfaces are used, but 99% of all rotating shafts, axes, etc. are made in the form of cylinders. In order to better understand what a surface of revolution is, we can consider how the cylinder itself is formed.
Let's say there is a certain straight line a, located vertically. ABCD is a rectangle, one of whose sides (segment AB) lies on a line a. If we rotate a rectangle around a straight line, as shown in the figure, the volume that it will occupy while rotating will be our body of revolution - a right circular cylinder with height H = AB = DC and radius R = AD = BC.
In this case, as a result of rotating the figure - a rectangle - a cylinder is obtained. By rotating a triangle, you can get a cone, by rotating a semicircle - a ball, etc.
Cylinder surface area
In order to calculate the surface area of an ordinary right circular cylinder, it is necessary to calculate the areas of the bases and lateral surfaces.
First, let's look at how the lateral surface area is calculated. This is the product of the circumference of the cylinder and the height of the cylinder. The circumference, in turn, is equal to twice the product of the universal number P by the radius of the circle.
The area of a circle is known to be equal to the product P per square radius. So, by adding the formulas for the area of determining the lateral surface with the double expression for the area of the base (there are two of them) and making simple algebraic transformations, we obtain the final expression for determining the surface area of the cylinder.
Determining the volume of a figure
The volume of a cylinder is determined according to the standard scheme: the surface area of the base is multiplied by the height.
Thus, the final formula looks like this: the desired value is defined as the product of the height of the body by the universal number P and by the square of the radius of the base.
The resulting formula, it must be said, is applicable to solving the most unexpected problems. In the same way as the volume of the cylinder, for example, the volume of electrical wiring is determined. This may be necessary to calculate the mass of the wires.
The only difference in the formula is that instead of the radius of one cylinder there is the diameter of the wiring strand divided in half and the number of strands in the wire appears in the expression N. Also, instead of height, the length of the wire is used. In this way, the volume of the “cylinder” is calculated not just by one, but by the number of wires in the braid.
Such calculations are often required in practice. After all Substantial part water tanks are made in the shape of a pipe. And it is often necessary to calculate the volume of a cylinder even in the household.
However, as already mentioned, the shape of the cylinder can be different. And in some cases it is necessary to calculate what the volume of an inclined cylinder is.
The difference is that the surface area of the base is not multiplied by the length of the generatrix, as in the case of a straight cylinder, but by the distance between the planes - a perpendicular segment constructed between them.
As can be seen from the figure, such a segment is equal to the product of the length of the generatrix and the sine of the angle of inclination of the generatrix to the plane.
How to build a cylinder development
In some cases, it is necessary to cut out a cylinder ream. The figure below shows the rules by which a blank is constructed for the manufacture of a cylinder with a given height and diameter.
Please note that the drawing is shown without seams.
Differences between a beveled cylinder
Let us imagine a certain straight cylinder, bounded on one side by a plane perpendicular to the generators. But the plane bounding the cylinder on the other side is not perpendicular to the generators and not parallel to the first plane.
The figure shows a beveled cylinder. Plane A at a certain angle, different from 90° to the generators, intersects the figure.
Such geometric shape more often found in practice in the form of pipeline connections (elbows). But there are even buildings built in the form of a beveled cylinder.
Geometric characteristics of a beveled cylinder
The tilt of one of the planes of a beveled cylinder slightly changes the procedure for calculating both the surface area of such a figure and its volume.
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 base and the side surface. Since the bases in the cylinder are equal, its total area will be calculated by the formula:
We will consider an example of calculating the area of a 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 will need to apply: 
We remember that in these calculations the constant number Π = 3.1415926 is used, which is calculated as the ratio of the circumference of a circle to its diameter. This number is a mathematical constant. We will also look at 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 let's look at a problem in which we need to calculate the total area of a cylinder. In the given figure, the height is h = 4 cm, r = 2 cm. Let us find the total area of the cylinder.
First, let's calculate the area of the bases:
Now let's look at an example of calculating the area of the lateral surface of a cylinder. When expanded, it represents a rectangle. Its area is calculated using the above formula. Let's substitute all the data into it:
The total area of a circle is the sum of double 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 axial cross-sectional area of a cylinder is derived from the calculation formula:
A cylinder is a symmetrical spatial figure, the properties of which are considered in high school in the course of stereometry. To describe it, linear characteristics such as height and base radius are used. In this article we will consider questions regarding what the axial section of a cylinder is and how to calculate its parameters through the basic linear characteristics of the figure.
Geometric figure
First, let's define the figure that will be discussed in the article. A cylinder is a surface formed by parallel movement of a segment of a fixed length along a certain curve. The main condition for this movement is that the segment should not belong to the plane of the curve.
The figure below shows a cylinder whose curve (guide) is an ellipse.
Here a segment of length h is its generator and height.
It can be seen that the cylinder consists of two identical bases (ellipses in this case), which lie in parallel planes, and a side surface. The latter belongs to all points of the forming lines.
Before moving on to considering the axial section of cylinders, we will tell you what types of these figures there are.
If the generating line is perpendicular to the bases of the figure, then we speak of a straight cylinder. Otherwise the cylinder will be inclined. If you connect the central points of two bases, the resulting straight line is called the axis of the figure. The figure below shows the difference between straight and inclined cylinders.
It can be seen that for a straight figure, the length of the generating segment coincides with the value of the height h. For an inclined cylinder, the height, that is, the distance between the bases, is always less than the length of the generatrix line.
Axial section of a straight cylinder
Axial is any section of the cylinder that contains its axis. This definition means that the axial section will always be parallel to the generatrix.
In a straight cylinder, the axis passes through the center of the circle and is perpendicular to its plane. This means that the circle under consideration will intersect along its diameter. The figure shows half a cylinder, which is the result of the intersection of the figure with a plane passing through the axis.
It is not difficult to understand that the axial section of a straight line round cylinder is a rectangle. Its sides are the diameter d of the base and the height h of the figure.
Let us write the formulas for the axial cross-sectional area of the cylinder and the length h d of its diagonal:
A rectangle has two diagonals, but both are equal to each other. If the radius of the base is known, then it is not difficult to rewrite these formulas through it, given that it is half the diameter.
Axial section of an inclined cylinder
The picture above shows a slanted cylinder made of paper. If you make its axial section, you will no longer get a rectangle, but a parallelogram. Its sides are known quantities. One of them, as in the case of the cross-section of a straight cylinder, is equal to the diameter d of the base, the other is the length of the forming segment. Let's denote it b.
To unambiguously determine the parameters of a parallelogram, it is not enough to know its side lengths. Another angle between them is needed. Let us assume that the acute angle between the guide and the base is α. This will also be the angle between the sides of the parallelogram. Then the formula for the axial cross-sectional area of an inclined cylinder can be written as follows:
The diagonals of the axial section of an inclined cylinder are somewhat more difficult to calculate. A parallelogram has two diagonals of different lengths. We present expressions without derivation that allow us to calculate the diagonals of a parallelogram using known sides and the acute angle between them:
l 1 = √(d 2 + b 2 - 2*b*d*cos(α));
l 2 = √(d 2 + b 2 + 2*b*d*cos(α))
Here l 1 and l 2 are the lengths of the small and large diagonals, respectively. These formulas can be obtained independently by considering each diagonal as a vector by entering rectangular system coordinates on the plane.
Straight Cylinder Problem
We will show you how to use the knowledge gained to solve the following problem. Let us be given a round straight cylinder. It is known that the axial cross section of a cylinder is square. What is the area of this section if the entire figure is 100 cm 2?
To calculate the required area, you need to find either the radius or the diameter of the base of the cylinder. To do this, we use the formula for total area S f figures:
Since the axial section is a square, this means that the radius r of the base is half the height h. Taking this into account, we can rewrite the equality above as:
S f = 2*pi*r*(r + 2*r) = 6*pi*r 2
Now we can express the radius r, we have:
Since the side square section is equal to the diameter of the base of the figure, then to calculate its area S the following formula will be valid:
S = (2*r) 2 = 4*r 2 = 2*S f / (3*pi)
We see that the required area is uniquely determined by the surface area of the cylinder. Substituting the data into equality, we come to the answer: S = 21.23 cm 2.
A cylinder is a figure consisting of a cylindrical surface and two circles located in parallel. Calculating the area of a cylinder is a problem in the geometric branch of mathematics, which can be solved quite simply. There are several methods for solving it, which in the end always come down to one formula.
How to find the area of a cylinder - calculation rules
- To find out the area of the cylinder, you need to add the two areas of the base with the area of the side surface: S = Sside + 2Sbase. In a more expanded 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 lying at its base are known. In this case, you can express the radius from the circumference, if given. The height can be found if the value of the generator is specified in the condition. In this case, the generatrix will be equal to the height. The formula for the lateral surface of this body looks like this: S= 2 π rh.
- The area of the base is calculated using 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 may be given. With this formula, the radius is expressed quite easily. С=2π r, r= С/2π. You must also remember that the radius is half the diameter.
- When performing all these calculations, the number π usually does not translate into 3.14159... It just needs to be added next to numerical value, which was obtained as a result of calculations.
- Next, you just need to multiply the found area of the base by 2 and add to the resulting number the calculated area of the lateral surface of the figure.
- If the problem indicates that the cylinder has an axial section and that it 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 lying at the base of the body. The length of the figure will be equal to the generatrix or height of the cylinder. It is necessary to calculate the required values and substitute them into the 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 lateral surface, the length is multiplied by two radii and 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 complicated in calculating the area of a cylinder. You just need to know the formulas and be able to derive from them the quantities necessary to carry out 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 as an example.
A cylinder has three surfaces: a top, a base, and a side surface.
The top and base 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 base of the cylinder) will be πr 2 + πr 2 = 2πr 2.
The third, side surface of the cylinder, is the curved wall of the cylinder. In order to better imagine 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 or bottom. Let's make a vertical cut on the side wall from the top to the bottom of the can (Step 1 in the figure) and try to open (straighten) the resulting figure as much as possible (Step 2).
After the resulting jar is fully opened, 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's return for a moment to the original cylinder. The vertex 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 received a formula for calculating the area of the lateral surface of the cylinder.
Formula for the lateral surface area of a cylinder
S side = 2πrh
Total surface area of a cylinder
Finally, if we add the area of all three surfaces, we get the formula for the total surface area of a cylinder. The surface area of a 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 identical to the formula 2πr (r + h).
Formula for the total surface area of a cylinder
S = 2πr 2 + 2πrh = 2πr(r + h)
r – radius of the cylinder, h – 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 using the formula: S side. = 2πrh
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