How to calculate the surface area of ​​a cylinder is the topic of this article. At any math problem you need to start with data entry, determine what is known and what to operate in the future, and only then proceed directly to the calculation.

This voluminous body is geometric figure cylindrical, 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 the axis being one of its sides.

It follows from this that the described curve above and below the cylinder will be a circle, the main indicator of which is the radius or diameter.

Cylinder Surface Area - Online Calculator

This function finally facilitates the calculation process, and everything comes down to automatic substitution of the given values ​​of 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 how the sweep looks 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 is the radius of the circle. The other side of the rectangle is equal to the height h. It won't be hard to find what you're looking for.

Sside= 2π *r*h,

where number π = 3.14.

Full surface area of ​​a cylinder

To find the total area of ​​the cylinder, you need to get S side add the areas of two circles, the top and bottom of the cylinder, which are calculated by the formula S o =2π*r2.

The final formula looks like this:

Sfloor\u003d 2π * r 2+ 2π*r*h.

Cylinder area - formula in terms of diameter

To facilitate calculations, it is sometimes necessary to make calculations through the diameter. For example, there is a piece of a hollow pipe of known diameter.

Without bothering with unnecessary calculations, we have a ready-made formula. Algebra for 5th grade 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 you need to insert a value r=d/2.

Examples of calculating the area of ​​a cylinder

Armed with knowledge, let's get down to practice.

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 in terms of the radius:

S floor \u003d 2 * 3.14 * 24 2 + 2 * 3.14 * 24 * 100 \u003d 3617.28 + 15072 \u003d 18689.28 (mm 2).

We translate into the usual m 2 and get 0.01868928, approximately 0.02 m 2.

Example 2 It is required to find out the area of ​​​​the inner surface of the furnace asbestos pipe, the walls of which are lined with refractory bricks.

The data are as follows: diameter 0.2 m; height 2 m. We use the formula through the diameter:

S floor \u003d 3.14 * 0.2 2 / 2 + 3.14 * 0.2 * 2 \u003d 0.0628 + 1.256 \u003d 1.3188 m 2.

Example 3 How to find out how much material is needed to sew a bag, r \u003d 1 m and a height of 1 m.

One moment, there is a formula:

S side \u003d 2 * 3.14 * 1 * 1 \u003d 6.28 m 2.

Conclusion

At the end of the article, the question arose: are all these calculations and translations of one value into another really necessary? Why is all this necessary and most importantly, for whom? But do not neglect and forget simple formulas from high school.

The world has stood and will stand on elementary knowledge, including mathematics. And, starting any important work, it is never superfluous to refresh the calculation data in memory, applying them in practice with great effect. Accuracy - the politeness of kings.

The area of ​​each base of the cylinder is π r 2 , the area of ​​both bases will be 2π r 2 (Fig.).

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 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 ​​the two bases of the cylinder to the area of ​​the lateral surface of the cylinder, we get the total surface area of ​​the cylinder

S full \u003d 2πRH + 2πR 2 \u003d 2πR (H + R).

Straight cylinder volume

Theorem. Volume straight cylinder is equal to the product the area of ​​its base to its height , i.e.

where Q is the base area and H is the height of the cylinder.

Since the area of ​​the base of the cylinder is Q, 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 sequences of prisms whose bases are the described and inscribed polygons considered above, and whose lateral edges are parallel to the generatrix of the given cylinder and have length H. These prisms are described 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 \u003d π R 2, and therefore

Cylinder (circular cylinder) - a body that consists of two circles combined by parallel transfer, and all segments connecting the corresponding points of these circles. The circles are called the bases of the cylinder, and the segments connecting the corresponding points of the circles of the circles are called the generators of the cylinder.

The bases of the cylinder are equal and lie in parallel planes, and the generators of the cylinder are parallel and equal. The surface of a cylinder consists of bases and a side surface. The lateral surface is formed by generators.

A cylinder is called straight if its generators are perpendicular to the planes of the base. A cylinder can be considered as a body obtained by rotating a rectangle around one of its sides as an axis. There are other types of cylinder - elliptical, hyperbolic, parabolic. A prism is also considered as a kind of cylinder.

Figure 2 shows an inclined cylinder. Circles with centers O and O 1 are its bases.

The radius of a cylinder is the radius of its base. The height of the cylinder is the distance between the planes of the bases. The axis of a cylinder is a straight line passing through the centers of the bases. It is parallel to the generators. The section of a cylinder by a plane passing through the axis of the cylinder is called an axial section. The plane passing through the generatrix of a straight cylinder and perpendicular to the axial section drawn through this generatrix is ​​called the tangent plane of the cylinder.

A plane perpendicular to the axis of the cylinder intersects it side surface around a circle equal to the circumference of the base.

A prism inscribed in a cylinder is a prism whose bases are equal polygons inscribed in the bases of the cylinder. Its lateral edges are generatrices of the cylinder. A prism is said to be circumscribed near a cylinder if its bases are equal polygons circumscribed near the bases of the cylinder. The planes of its faces touch the side surface of the cylinder.

The area of ​​the lateral surface of the cylinder can be calculated by multiplying the length of the generatrix by the perimeter of the section of the cylinder by a plane perpendicular to the generatrix.

The lateral surface area of ​​a right cylinder can be found from its development. The development of the cylinder is a rectangle with height h and length P, which is equal to the perimeter of the base. Therefore, the area of ​​the lateral surface of the cylinder is equal to the area of ​​its development and is calculated by the formula:

In particular, for a right circular cylinder:

P = 2πR, and Sb = 2πRh.

The total surface area of ​​a cylinder is equal to the sum of the areas of its lateral surface and its bases.

For a straight circular cylinder:

S p = 2πRh + 2πR 2 = 2πR(h + R)

There are two formulas for finding the volume of an inclined cylinder.

You can find the volume by multiplying the length of the generatrix by the cross-sectional area of ​​\u200b\u200bthe cylinder by a plane perpendicular to the generatrix.

The volume of an inclined cylinder is equal to the product of the area of ​​the base and the height (the distance between the planes in which the bases lie):

V = Sh = S l sin α,

where l is the length of the generatrix, and α is the angle between the generatrix and the plane of the base. For a straight cylinder h = l.

The formula for finding the volume of a circular cylinder is as follows:

V \u003d π R 2 h \u003d π (d 2 / 4) h,

where d is the base diameter.

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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 equal square root from 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 cross section passes 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 generatrices 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.

V 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.