What is the lateral surface area of a cylinder. Axial section of a straight and inclined cylinder
There are many tasks associated with a cylinder. They need to find the radius and height of the body or the type of its section. Plus, sometimes you need to calculate the area of a cylinder and its volume.
Which body is a cylinder?
I know school curriculum the circular, that is, being such at the base, cylinder is studied. But they also highlight the elliptical appearance of this figure. From the name it is clear that its base will be an ellipse or oval.
The cylinder has two bases. They are equal to each other and are connected by line segments that match the corresponding base points. They are called generatrices of the cylinder. All generators are parallel to each other and equal. They are the ones that make up the lateral surface of the body.
In general, a cylinder is an inclined body. If the generators make a right angle with the bases, then they are already talking about a straight figure.
Interestingly, a circular cylinder is a body of revolution. It is obtained by rotating a rectangle around one of its sides.
The main elements of the cylinder
The main elements of the cylinder are as follows.
- Height. It is the shortest distance between the bases of the cylinder. If it is straight, then the height coincides with the generatrix.
- Radius. The same as the one that can be drawn at the base.
- Axis. It is a straight line that contains the centers of both bases. The axis is always parallel to all generators. In a straight cylinder, it is perpendicular to the bases.
- Axial section. It is formed when the plane that contains the axis intersects the cylinder.
- Tangent plane. It passes through one of the generatrices and is perpendicular to the axial section, which is drawn through this generatrix.
How is the cylinder connected with a prism inscribed in it or described around it?
Sometimes there are problems in which it is necessary to calculate the area of a cylinder, and some elements of the prism associated with it are known. How do these figures relate?
If a prism is inscribed in a cylinder, then its bases are equal polygons. Moreover, they are inscribed in the corresponding cylinder bases. The lateral edges of the prism coincide with the generatrices.
In the described prism, the bases are regular polygons... They are described around the circles of the cylinder, which are its bases. The planes that contain the faces of the prism touch the cylinder along the generatrix.
About the area of the lateral surface and the base for a straight circular cylinder
If you unroll the side surface, you get a rectangle. Its sides will coincide with the generatrix and the circumference of the base. Therefore, the lateral area of the cylinder will be equal to the product of these two values. If you write down the formula, you get the following:
S side = l * n,
where n is the generator, l is the circumference.
Moreover, the last parameter is calculated by the formula:
l = 2 π * r,
here r is the radius of the circle, π is the number "pi" equal to 3.14.
Since the base is a circle, its area is calculated using the following expression:
S main = π * r 2.
About the area of the entire surface of a straight circular cylinder
Since it is formed by two bases and a lateral surface, you need to add these three values. That is, the total area of the cylinder will be calculated by the formula:
S floor = 2 π * r * n + 2 π * r 2.
Often it is written in a different form:
S floor = 2 π * r (n + r).
About the areas of an inclined circular cylinder
As for the foundations, all the formulas are the same, because they are still circles. And here side surface no longer gives a rectangle.
To calculate the area of the lateral surface of an inclined cylinder, you will need to multiply the values of the generatrix and the perimeter of the section, which will be perpendicular to the selected generatrix.
The formula looks like this:
S side = x * P,
where x is the length of the generatrix of the cylinder, P is the perimeter of the section.
By the way, it is better to choose a section so that it forms an ellipse. Then the calculations of its perimeter will be simplified. The length of the ellipse is calculated using a formula that gives an approximate answer. But it is often enough for the tasks of the school course:
l = π * (a + b),
where "a" and "b" are the semiaxes of the ellipse, that is, the distance from the center to its nearest and farthest points.
The area of the entire surface must be calculated using the following expression:
S floor = 2 π * r 2 + x * R.
What are some sections of a right circular cylinder equal to?
When the section passes through the axis, then its area is determined as the product of the generatrix and the diameter of the base. This is due to the fact that it looks like a rectangle, the sides of which coincide with the designated elements.
To find the cross-sectional area of a cylinder that is parallel to the axial one, you will also need a formula for a rectangle. In this situation, one side of it will still coincide with the height, while the other is equal to the chord of the base. The latter coincides with the section line at the base.
When the section is perpendicular to the axis, then it looks like a circle. Moreover, its area is the same as at the base of the figure.
An intersection at a certain angle to the axis is also possible. Then, in the section, an oval or part of it is obtained.
Examples of tasks
Task number 1. Given a straight cylinder, the base area of which is 12.56 cm 2. It is necessary to calculate the total area of the cylinder if its height is 3 cm.
Solution. It is necessary to use the formula for the full area of the circular straight cylinder... But it lacks data, namely the base radius. But the area of the circle is known. It is easy to calculate the radius from it.
It turns out to be equal to the square root of the quotient, which is obtained by dividing the area of the base by pi. After dividing 12.56 by 3.14, 4 comes out. Square root of 4 is 2. Therefore, the radius will have exactly this value.
Answer: S floor = 50.24 cm 2.
Task number 2. A cylinder with a radius of 5 cm is intercepted by a plane parallel to the axis. The distance from the section to the axis is 3 cm. The height of the cylinder is 4 cm. It is required to find the section area.
Solution. Sectional shape - rectangular. One side of it coincides with the height of the cylinder, and the other is equal to the chord. If the first value is known, then the second must be found.
For this, an additional construction should be made. Draw two segments at the base. Both of them will start at the center of the circle. The first will end at the center of the chord and equal the known distance to the axis. The second is at the end of the chord.
You will get a right-angled triangle. The hypotenuse and one of the legs are known in it. The hypotenuse matches the radius. The second leg is equal to half of the chord. The unknown leg, multiplied by 2, will give the desired chord length. Let's calculate its value.
In order to find the unknown leg, you need to square the hypotenuse and the known leg, subtract the second from the first and extract the square root. The squares are 25 and 9. Their difference is 16. After extracting the square root, 4 remains. This is the desired leg.
The chord will be 4 * 2 = 8 (cm). Now you can calculate the cross-sectional area: 8 * 4 = 32 (cm 2).
Answer: S section is equal to 32 cm 2.
Task number 3. It is necessary to calculate the area of the axial section of the cylinder. It is known that a cube with an edge of 10 cm is inscribed in it.
Solution. The axial section of the cylinder coincides with the rectangle that passes through the four vertices of the cube and contains the diagonals of its bases. The side of the cube is the generatrix of the cylinder, and the diagonal of the base coincides with the diameter. The product of these two values will give the area that you need to know in the problem.
To find the diameter, you need to use the knowledge that at the base of the cube is a square, and its diagonal forms an equilateral right triangle... Its hypotenuse is the required figure diagonal.
To calculate it, you need the formula of the Pythagorean theorem. You need to square the side of the cube, multiply it by 2 and extract the square root. Ten to the second degree is one hundred. Multiplied by 2 - two hundred. The square root of 200 is 10√2.
The section is again a rectangle with sides 10 and 10√2. Its area can be easily calculated by multiplying these values.
Answer. S section = 100√2 cm 2.
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 the height and radius of the base are used. In this article, we will consider questions regarding what is the axial section of a cylinder, and how to calculate its parameters through the basic linear characteristics of the figure.
Geometric figure
First, let's define the shape that will be discussed in the article. A cylinder is a surface formed by parallel displacement of a segment of a fixed length along a certain curve. The main condition for this movement is that the segment of the plane of the curve should not belong.
The figure below shows a cylinder whose curve (guide) is an ellipse.
Here the segment of length h is its generatrix 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 lateral surface. All points of the generating lines belong to the latter.
Before proceeding to the consideration of the axial section of the cylinders, we will tell you what types of these figures 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 tilted. If you connect the center points of the two bases, then the resulting straight line is called the axis of the figure. The figure below demonstrates the difference between straight and tilted 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 generating line.
Axial section of a straight cylinder
Axial is any section of a cylinder that contains its axis. This definition means that the axial section will always be parallel to the generatrix line.
In a cylinder, the straight axis passes through the center of the circle and is perpendicular to its plane. This means that the circle under consideration will intersect in its diameter. The figure shows the half of the 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 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 area of the axial section of the cylinder and the length h d of its diagonal:
The rectangle has two diagonals, but they are both 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 tilted cylinder made of paper. If you make its axial section, you will get not a rectangle, but a parallelogram. Its sides are known quantities. One of them, as in the case of the section of a straight cylinder, is equal to the diameter d of the base, while the other is the length of the generating segment. We denote it by b.
For an unambiguous determination of the parallelogram parameters, it is not enough to know its side lengths. An angle between them is also required. Suppose the acute angle between the rail and the base is α. It will also be the angle between the sides of the parallelogram. Then the formula for the axial 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. Let us present, without derivation, expressions that allow us to calculate the diagonals of the parallelogram along the known sides and sharp corner 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 if we consider each diagonal as a vector, introducing a rectangular coordinate system on the plane.
Straight Cylinder Problem
Let's show how to use the knowledge gained to solve the following problem. Let a round straight cylinder be given. It is known that the axial section of a cylinder is a square. What is the area of this section if the whole 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 the total area S f of the figure:
Since the axial section is a square, this means that the radius r of the base is twice less height h. With this in mind, we can rewrite the above equality 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 of the square section is equal to the diameter of the base of the figure, the following formula will be valid to calculate its area S:
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 geometric body bounded by two parallel planes and a cylindrical surface. In this article, we'll 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 a top hat is an ordinary tin which has no top cover and 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 open, 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 * 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
Cylinder (comes from Greek, from the words "roller", "roller") is a geometric body, which is bounded outside 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 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. There are several types of cylinders:
1. Circular, straight cylinder. For such a cylinder, the base and the guide are perpendicular to the generatrix line, and there is
2. Inclined cylinder. Its angle between the generating line and the base is not right.
3. Cylinder of a different shape. Hyperbolic, elliptical, parabolic and others.
The area of the 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 a cylinder for a circular, straight cylinder:
Sp = 2p Rh + 2p R2 = 2p R (h + R).
The area of the lateral surface is found a little more difficult than the area of the cylinder as a whole; it is calculated by multiplying the length of the generating line by the perimeter of the section formed by the plane, which is perpendicular to the generating line.
A given cylinder for a circular, straight cylinder is recognized by the unfolding of this object.
A flat pattern is a rectangle that has a height h and a length P that equals 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 should be equal to the product of the length of its generatrix line and the perimeter of the section, which is perpendicular to this generatrix 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 the 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 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 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 build a circle at the base of the cylinder, draw a radius and plot 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 touch 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 a circle.
The area of the circle is:
S env. = 2п R2.
To find R, you need to divide its length C by 2n:
R \ u003d C \ 2п, where n is the number pi, a mathematical constant calculated to work with the data of the circle and equal to 3.14.
It is a geometric body bounded by two parallel planes and a cylindrical surface.
The cylinder consists of a lateral surface and two bases. The cylinder surface area formula includes a separate calculation for base and side area. 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 learn all the necessary formulas. First, we need a 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 in these calculations a 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 consider an example of calculating the area of the base of a cylinder a little later.
Cylinder lateral surface area
The formula for the lateral surface area of a cylinder is the product of the base length by its height:
Now let's 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 us find the total area of the cylinder.
First, let's calculate the area of the bases:
Now let's consider an example of calculating the area of the lateral surface of a cylinder. When expanded, it is 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 area of the axial section of a cylinder is derived from the calculation formula:
