The bodies of rotation studied at school are the cylinder, cone and ball.
If in a problem on the Unified State Exam in mathematics you need to calculate the volume of a cone or the area of a sphere, consider yourself lucky.
Apply formulas for volume and surface area of a cylinder, cone and sphere. All of them are in our table. Learn by heart. This is where knowledge of stereometry begins.
Sometimes it's good to draw the view from above. Or, as in this problem, from below.
2. How many times is the volume of a cone circumscribed about a regular quadrangular pyramid greater than the volume of a cone inscribed in this pyramid?
It's simple - draw the view from below. We see that the radius of the larger circle is times larger than the radius of the smaller one. The heights of both cones are the same. Therefore, the volume of the larger cone will be twice as large.
Another important point. We remember that in the problems of part B of the Unified State Examination in mathematics, the answer is written as an integer or a final decimal fraction. Therefore, there should not be any or in your answer in part B. There is no need to substitute the approximate value of the number either! It must definitely shrink! It is for this purpose that in some problems the task is formulated, for example, as follows: “Find the area of the lateral surface of the cylinder divided by.”
Where else are the formulas for volume and surface area of bodies of revolution used? Of course, in problem C2 (16). We will also tell you about it.
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
S = 226.08 + 150.72
The surface area of the cylinder is 376.8.
There are a large number of problems associated with the cylinder. In them you 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?
In the school curriculum, a circular cylinder, that is, one at the base, is studied. But the elliptical appearance of this figure is also distinguished. From the name it is clear that its base will be an ellipse or an oval.
The cylinder has two bases. They are equal to each other and are connected by segments that combine the corresponding points of the bases. They are called the generators of the cylinder. All generators are parallel to each other and equal. They 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 we speak of a straight figure.
Interestingly, a circular cylinder is a body of revolution. It is obtained by rotating a rectangle around one of its sides.
Main elements of the cylinder
The main elements of the cylinder look like this.
- Height. It is the shortest distance between the bases of the cylinder. If it is straight, then the height coincides with the generatrix.
- Radius. Coincides with the one that can be drawn at the base.
- Axis. This 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 a cylinder intersects a plane containing an axis.
- Tangent plane. It passes through one of the generatrices and is perpendicular to the axial section, which is drawn through this generatrix.
How is a cylinder connected to a prism inscribed in it or described around it?
Sometimes there are problems in which you need to calculate the area of a cylinder, but some elements of the associated prism 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 bases of the cylinder. The lateral edges of the prism coincide with the generators.
The described prism has regular polygons at its base. 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 their generators.
On the area of the lateral surface and base for a right circular cylinder
If you unwrap the side surface, you will 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 quantities. 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 using 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 .
On the area of the entire surface of a right circular cylinder
Since it is formed by two bases and a side surface, you need to add these three quantities. That is, the total area of the cylinder will be calculated by the formula:
S floor = 2 π * r * n + 2 π * r 2 .
It is often written in a different form:
S floor = 2 π * r (n + r).
On the areas of an inclined circular cylinder
As for the bases, all the formulas are the same, because they are still circles. But the 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 cylinder generatrix, P is the perimeter of the section.
By the way, it is better to choose a section such 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 sufficient for the tasks of a school course:
l = π * (a + b),
where “a” and “b” are the semi-axes 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?
When a section passes through an axis, its area is determined as the product of the generatrix and the diameter of the base. This is explained by the fact that it has the shape of 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 of its sides will still coincide with the height, and the other will be equal to the chord of the base. The latter coincides with the section line along the base.
When the section is perpendicular to the axis, it looks like a circle. Moreover, its area is the same as that of the base of the figure.
It is also possible to intersect at some angle to the axis. Then the cross-section results in an oval or part of it.
Sample problems
Task No. 1. Given a straight cylinder whose base area 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 total area of a circular straight cylinder. But it lacks data, namely the radius of the base. But the area of the circle is known. It is easy to calculate the radius from this.
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, the result is 4. The square root of 4 is 2. Therefore, the radius will have this value.
Answer: S floor = 50.24 cm 2.
Task No. 2. A cylinder with a radius of 5 cm is cut 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. You need to find the cross-sectional area.
Solution. The cross-sectional shape is rectangular. One of its sides coincides with the height of the cylinder, and the other is equal to the chord. If the first quantity is known, then the second one needs to be found.
To do this, additional construction must be made. At the base we draw two segments. They will both start at the center of the circle. The first will end at the center of the chord and equal to the known distance to the axis. The second is at the end of the chord.
You will get a right triangle. The hypotenuse and one of the legs are known in it. The hypotenuse coincides with the radius. The second leg is equal to half 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 will need to square the hypotenuse and the known leg, subtract the second from the first and take the square root. The squares are 25 and 9. Their difference is 16. After taking the square root, 4 remains. This is the desired leg.
The chord will be equal to 4 * 2 = 8 (cm). Now you can calculate the cross-sectional area: 8 * 4 = 32 (cm 2).
Answer: S cross is equal to 32 cm 2.
Task No. 3. It is necessary to calculate the axial cross-sectional area 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 a 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 quantities will give the area that you need to find out in the problem.
To find the diameter, you will need to use the knowledge that the base of the cube is a square, and its diagonal forms an equilateral right triangle. Its hypotenuse is the desired diagonal of the figure.
To calculate it, you will need the formula of the Pythagorean theorem. You need to square the side of the cube, multiply it by 2 and take the square root. Ten to the second power is one hundred. Multiplied by 2 is 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.
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 a cylindrical geometric figure, bounded at the top and bottom 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 the total area of the cylinder, you need to use the resulting 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 a 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 you need to insert the value into the full formula 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.
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