The area of the lateral surface of a cylinder is equal to the area of a rectangle whose base is 2π r, and the height is equal to the height of the cylinder h, i.e. 2π rh.
The total surface of the cylinder will be: 2π r 2 + 2π rh= 2π r(r+ h).
The area of the lateral surface of the cylinder is taken to be sweep area its lateral surface.
Therefore, the area of the lateral surface of a right circular cylinder is equal to the area of the corresponding rectangle (Fig.) and is calculated by the formula
S b.c. = 2πRH, (1)
If we add the area of its two bases to the area of the lateral surface of the cylinder, we obtain the total surface area of the cylinder
S full =2πRH + 2πR 2 = 2πR (H + R).
Volume of a straight cylinder
Theorem. The volume of a straight cylinder is equal to the product of the area of its base and its height , i.e.where Q is the area of the base, and H is the height of the cylinder.
Since the area of the base of the cylinder is Q, then there are sequences of circumscribed and inscribed polygons with areas Q n and Q' n such that
\(\lim_(n \rightarrow \infty)\) Q n= \(\lim_(n \rightarrow \infty)\) Q’ n= Q.
Let us construct a sequence of prisms whose bases are the described and inscribed polygons discussed above, and whose side edges are parallel to the generatrix of the given cylinder and have length H. These prisms are circumscribed and inscribed for the given cylinder. Their volumes are found by the formulas
V n= Q n H and V' n= Q' n H.
Hence,
V= \(\lim_(n \rightarrow \infty)\) Q n H = \(\lim_(n \rightarrow \infty)\) Q’ n H = QH.
Consequence.
The volume of a right circular cylinder is calculated by the formula
V = π R 2 H
where R is the radius of the base and H is the height of the cylinder.
Since the base of a circular cylinder is a circle of radius R, then Q = π R 2, and therefore
Cylinder (comes from Greek language, from the words “roller”, “roller”) is a geometric body that is limited externally by a surface called cylindrical 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 formed by a straight line in space. These movements are such that the selected point of this straight line moves along a plane 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. There are several types of cylinders:
1. Circular, straight cylinder. Such a cylinder has a base and guide perpendicular to the generating line, and there is
2. Inclined cylinder. Its angle between the generating line and the base is not straight.
3. A cylinder of a different shape. Hyperbolic, elliptic, 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 the lateral surface.
The formula for calculating the total area of the cylinder for a circular, straight cylinder:
Sp = 2p Rh + 2p R2 = 2p R (h+R).
The area of the lateral surface is found to be a little more complicated than the area of the entire cylinder; it is calculated by multiplying the length of the generatrix line by the perimeter of the section formed by a plane that is perpendicular to the generatrix line.
The given cylinder for a circular, straight cylinder is recognized by the development of this object.
A development is a rectangle that has a height h and a length P, which is equal to the perimeter of the base.
It follows that lateral area cylinder is equal to the sweep area 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 area of the lateral surface should be equal to the product of the length of its generating line and the perimeter of the section, which is perpendicular to this generating line.
Unfortunately, there is no simple formula for expressing the lateral surface area of an inclined cylinder in terms of its height and the parameters of its base.
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, accordingly, 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 a given figure and 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 this section will be equal to the product of the height of this cylinder and a certain chord. To get a chord, you need to construct a circle at the base of the cylinder, draw a radius and plot 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 sought by 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 this section is found as the area of the circle.
The area of the circle is:
S env. = 2п R2.
To find R, you need to divide its length C by 2n:
R = C\2n, where n is pi, a mathematical constant calculated to work with circle data and equal to 3.14.
How to calculate the surface area of a cylinder is the topic of this article. In any mathematical problem, you need to start by entering data, determine what is known and what to operate with in the future, and only then proceed directly to the calculation.
This volumetric body is geometric figure cylindrical in shape, bounded above and below by two parallel planes. If you apply a little imagination, you will notice that a geometric body is formed by rotating a rectangle around an axis, with one of its sides being the axis.
It follows that the curve described above and below the cylinder will be a circle, the main indicator of which is the radius or diameter.
Surface area of a cylinder - online calculator
This function finally simplifies the calculation process, and it all comes down to automatically substituting the specified values for the height and radius (diameter) of the base of the figure. The only thing that is required is to accurately determine the data and not make mistakes when entering numbers.
Cylinder side surface area
First you need to imagine what a scan looks like in two-dimensional space.
This is nothing more than a rectangle, one side of which is equal to the circumference. Its formula has been known since time immemorial - 2π*r, Where r- radius of the circle. The other side of the rectangle is equal to the height h. Finding what you are looking for will not be difficult.
Sside= 2π *r*h,
where is the number π = 3.14.
Total surface area of a cylinder
To find full area cylinder needed to the received S side add the areas of two circles, the top and bottom of the cylinder, which are calculated using the formula S o =2π * r 2 .
The final formula looks like this:
Sfloor= 2π * r 2+ 2π * r * h.
Area of a cylinder - formula through diameter
To facilitate calculations, it is sometimes necessary to perform calculations through the diameter. For example, there is a piece of hollow pipe of known diameter.
Without bothering ourselves with unnecessary calculations, we have ready-made formula. 5th grade algebra comes to the rescue.
Sgender = 2π*r 2 + 2 π * r * h= 2 π*d 2 /4 + 2 π*h*d/2 = π *d 2 /2 + π *d*h,
Instead of r V full formula need to insert value r =d/2.
Examples of calculating the area of a cylinder
Armed with knowledge, let's start practicing.
Example 1. It is necessary to calculate the area of a truncated piece of pipe, that is, a cylinder.
We have r = 24 mm, h = 100 mm. You need to use the formula through the radius:
S floor = 2 * 3.14 * 24 2 + 2 * 3.14 * 24 * 100 = 3617.28 + 15072 = 18689.28 (mm 2).
We convert to the usual m2 and get 0.01868928, approximately 0.02 m2.
Example 2. It is required to find out the area of the internal surface of an asbestos stove pipe, the walls of which are lined with refractory bricks.
The data is as follows: diameter 0.2 m; height 2 m. We use the formula in terms of diameter:
S floor = 3.14 * 0.2 2 /2 + 3.14 * 0.2 * 2 = 0.0628 + 1.256 = 1.3188 m2.
Example 3. How to find out how much material is needed to sew a bag, r = 1 m and 1 m high.
One moment, there is a formula:
S side = 2 * 3.14 * 1 * 1 = 6.28 m2.
Conclusion
At the end of the article, the question arose: are all these calculations and conversions of one value to another really necessary? Why is all this needed and most importantly, for whom? But don’t neglect and forget simple formulas from high school.
The world has stood and will stand on elementary knowledge, including mathematics. And, when starting any important work, it is never a bad idea to refresh your memory of these calculations, applying them in practice with great effect. Accuracy - the politeness of kings.
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 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 lying at its base are known. IN in this case one can express the radius from the circumference of a circle, 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. Lateral surface formula given 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 is usually not translated into 3.14159... It just needs to be added next to the numerical value that was obtained as a result of the 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.
Cylinder radius formula:
where V is the volume of the cylinder, h is the height
A cylinder is a geometric body that is obtained by rotating a rectangle around its side. Also, a cylinder is a body bounded by a cylindrical surface and two parallel planes intersecting it. This surface is formed when a straight line moves parallel to itself. In this case, the selected point of the straight line moves along a certain plane curve (guide). This straight line is called the generator of the cylindrical surface.
Cylinder radius formula:
where Sb is the lateral surface area, h is the height
A cylinder is a geometric body that is obtained by rotating a rectangle around its side. Also, a cylinder is a body bounded by a cylindrical surface and two parallel planes intersecting it. This surface is formed when a straight line moves parallel to itself. In this case, the selected point of the straight line moves along a certain plane curve (guide). This straight line is called the generator of the cylindrical surface.
Cylinder radius formula:
where S is the total surface area, h is the height
