What is the lateral surface area. The area of ​​the lateral and full surface of the cone

- this is a figure, at the base of which lies an arbitrary polygon, and the side faces are represented by triangles. Their vertices lie at one point and correspond to the top of the pyramid.

The pyramid can be varied - triangular, quadrangular, hexagonal, etc. Its name can be determined depending on the number of corners adjacent to the base.
Correct pyramid called a pyramid, in which the sides of the base, angles, and edges are equal. Also, in such a pyramid, the area of ​​\u200b\u200bthe side faces will be equal.
The formula for the area of ​​the lateral surface of a pyramid is the sum of the areas of all its faces:
That is, to calculate the area of ​​the lateral surface of an arbitrary pyramid, it is necessary to find the area of ​​each individual triangle and add them together. If the pyramid is truncated, then its faces are represented by trapezoids. For the correct pyramid, there is another formula. In it, the lateral surface area is calculated through the semiperimeter of the base and the length of the apothem:

Consider an example of calculating the area of ​​the lateral surface of a pyramid.
Let a regular quadrangular pyramid be given. Base side b= 6 cm, and apothem a\u003d 8 cm. Find the area of ​​\u200b\u200bthe lateral surface.

At the base of a regular quadrangular pyramid lies a square. First, let's find its perimeter:

Now we can calculate the area of ​​the lateral surface of our pyramid:

To find the total area of ​​a polyhedron, you need to find the area of ​​its base. The formula for the area of ​​the base of a pyramid may differ, depending on which polygon lies at the base. To do this, use the formula for the area of ​​a triangle, parallelogram area etc.

Consider an example of calculating the area of ​​\u200b\u200bthe base of the pyramid given by our conditions. Since the pyramid is regular, it has a square at its base.
square area calculated by the formula: ,
where a is the side of the square. We have it equal to 6 cm. So the area of ​​\u200b\u200bthe base of the pyramid:

Now it remains only to find the total area of ​​the polyhedron. The formula for the area of ​​a pyramid is the sum of the area of ​​its base and its lateral surface.

The area of ​​the lateral surface of an arbitrary pyramid is equal to the sum of the areas of its lateral faces. It makes sense to give a special formula for expressing this area in the case of a regular pyramid. So, let a regular pyramid be given, at the base of which lies a regular n-gon with a side equal to a. Let h be the height of the side face, also called apothema pyramids. The area of ​​one side face is 1/2ah, and the entire side surface of the pyramid has an area equal to n/2ha. Since na is the perimeter of the base of the pyramid, we can write the found formula as follows:

Lateral surface area of a regular pyramid is equal to the product of its apothem by half the perimeter of the base.

Concerning total surface area, then simply add the area of ​​\u200b\u200bthe base to the side.

Inscribed and circumscribed sphere and ball. It should be noted that the center of the sphere inscribed in the pyramid lies at the intersection of the bisector planes of the internal dihedral angles of the pyramid. The center of the sphere described near the pyramid lies at the intersection of planes passing through the midpoints of the edges of the pyramid and perpendicular to them.

Truncated pyramid. If the pyramid is cut by a plane parallel to its base, then the part enclosed between the cutting plane and the base is called truncated pyramid. The figure shows a pyramid, discarding its part lying above the cutting plane, we get a truncated pyramid. It is clear that the small pyramid to be discarded is homothetic to the large pyramid with the center of the homothety at the apex. The similarity coefficient is equal to the ratio of heights: k=h 2 /h 1 , or side ribs, or other corresponding linear dimensions of both pyramids. We know that the areas of similar figures are related as squares of linear dimensions; so the areas of the bases of both pyramids (i.e. spare the bases of the truncated pyramid) are related as

Here S 1 is the area of ​​the lower base, and S 2 is the area of ​​the upper base of the truncated pyramid. The side surfaces of the pyramids are in the same ratio. There is a similar rule for volumes.

Volumes of similar bodies are related as cubes of their linear dimensions; for example, the volumes of the pyramids are related as the products of their heights by the area of ​​​​the bases, from which our rule immediately follows. It has a completely general character and directly follows from the fact that the volume always has the dimension of the third power of length. Using this rule, we derive a formula expressing the volume of a truncated pyramid in terms of the height and areas of the bases.

Let a truncated pyramid with height h and base areas S 1 and S 2 be given. If we imagine that it is extended to the full pyramid, then the similarity coefficient of the full pyramid and the small pyramid can be easily found as the root of the ratio S 2 /S 1. The height of the truncated pyramid is expressed as h = h 1 - h 2 = h 1 (1 - k). Now we have for the volume of the truncated pyramid (V 1 and V 2 denote the volumes of the full and small pyramids)

truncated pyramid volume formula

We derive the formula for the area S of the lateral surface of a regular truncated pyramid through the perimeters P 1 and P 2 of the bases and the length of the apothem a. We argue in exactly the same way as when deriving the formula for volume. We supplement the pyramid with the upper part, we have P 2 \u003d kP 1, S 2 \u003d k 2 S 1, where k is the similarity coefficient, P 1 and P 2 are the perimeters of the bases, and S 1 and S 2 are the horses of the side surfaces of the entire resulting pyramid and its top, respectively. For the lateral surface we find (a 1 and a 2 - apothems of the pyramids, a \u003d a 1 - a 2 \u003d a 1 (1-k))

formula for the lateral surface area of ​​a regular truncated pyramid

When preparing for the exam in mathematics, students have to systematize their knowledge of algebra and geometry. I would like to combine all known information, for example, how to calculate the area of ​​a pyramid. Moreover, starting from the base and side faces to the entire surface area. If the situation is clear with the side faces, since they are triangles, then the base is always different.

What to do when finding the area of ​​the base of the pyramid?

It can be absolutely any figure: from an arbitrary triangle to an n-gon. And this base, in addition to the difference in the number of angles, can be a regular figure or an incorrect one. In the USE tasks of interest to schoolchildren, there are only tasks with the correct figures at the base. Therefore, we will only talk about them.

right triangle

That is equilateral. One in which all sides are equal and denoted by the letter "a". In this case, the area of ​​\u200b\u200bthe base of the pyramid is calculated by the formula:

S = (a 2 * √3) / 4.

Square

The formula for calculating its area is the simplest, here "a" is the side again:

Arbitrary regular n-gon

The side of a polygon has the same designation. For the number of corners, the Latin letter n is used.

S = (n * a 2) / (4 * tg (180º/n)).

How to proceed when calculating the lateral and total surface area?

Since the base is a regular figure, all the faces of the pyramid are equal. Moreover, each of them is an isosceles triangle, since the side edges are equal. Then, in order to calculate the lateral area of ​​\u200b\u200bthe pyramid, you need a formula consisting of the sum of identical monomials. The number of terms is determined by the number of sides of the base.

The area of ​​an isosceles triangle is calculated by the formula in which half the product of the base is multiplied by the height. This height in the pyramid is called apothem. Its designation is "A". The general formula for lateral surface area is:

S \u003d ½ P * A, where P is the perimeter of the base of the pyramid.

There are situations when the sides of the base are not known, but the side edges (c) and the flat angle at its vertex (α) are given. Then it is supposed to use such a formula to calculate the lateral area of ​​\u200b\u200bthe pyramid:

S = n/2 * in 2 sin α .

Task #1

Condition. Find the total area of ​​the pyramid if its base lies with a side of 4 cm, and the apothem has a value of √3 cm.

Solution. You need to start by calculating the perimeter of the base. Since this is a regular triangle, then P \u003d 3 * 4 \u003d 12 cm. Since the apothem is known, you can immediately calculate the area of ​​\u200b\u200bthe entire lateral surface: ½ * 12 * √3 = 6√3 cm 2.

For a triangle at the base, the following area value will be obtained: (4 2 * √3) / 4 \u003d 4√3 cm 2.

To determine the entire area, you will need to add the two resulting values: 6√3 + 4√3 = 10√3 cm 2.

Answer. 10√3 cm2.

Task #2

Condition. There is a regular quadrangular pyramid. The length of the side of the base is 7 mm, the side edge is 16 mm. You need to know its surface area.

Solution. Since the polyhedron is quadrangular and regular, then its base is a square. Having learned the areas of the base and side faces, it will be possible to calculate the area of ​​\u200b\u200bthe pyramid. The formula for the square is given above. And at the side faces, all sides of the triangle are known. Therefore, you can use Heron's formula to calculate their areas.

The first calculations are simple and lead to this number: 49 mm 2. For the second value, you will need to calculate the semi-perimeter: (7 + 16 * 2): 2 = 19.5 mm. Now you can calculate the area of ​​an isosceles triangle: √ (19.5 * (19.5-7) * (19.5-16) 2) = √2985.9375 = 54.644 mm 2. There are only four such triangles, so when calculating the final number, you will need to multiply it by 4.

It turns out: 49 + 4 * 54.644 \u003d 267.576 mm 2.

Answer. The desired value is 267.576 mm 2.

Task #3

Condition. For a regular quadrangular pyramid, you need to calculate the area. In it, the side of the square is 6 cm and the height is 4 cm.

Solution. The easiest way is to use the formula with the product of the perimeter and the apothem. The first value is easy to find. The second is a little more difficult.

We'll have to remember the Pythagorean theorem and consider It is formed by the height of the pyramid and the apothem, which is the hypotenuse. The second leg is equal to half the side of the square, since the height of the polyhedron falls into its middle.

The desired apothem (the hypotenuse of a right triangle) is √(3 2 + 4 2) = 5 (cm).

Now you can calculate the desired value: ½ * (4 * 6) * 5 + 6 2 \u003d 96 (cm 2).

Answer. 96 cm2.

Task #4

Condition. The correct side of its base is 22 mm, the side ribs are 61 mm. What is the area of ​​the lateral surface of this polyhedron?

Solution. The reasoning in it is the same as described in problem No. 2. Only there was given a pyramid with a square at the base, and now it is a hexagon.

First of all, the area of ​​\u200b\u200bthe base is calculated using the above formula: (6 * 22 2) / (4 * tg (180º / 6)) \u003d 726 / (tg30º) \u003d 726√3 cm 2.

Now you need to find out the semi-perimeter of an isosceles triangle, which is a lateral face. (22 + 61 * 2): 2 = 72 cm. It remains to calculate the area of ​​\u200b\u200beach such triangle using the Heron formula, and then multiply it by six and add it to the one that turned out for the base.

Calculations using the Heron formula: √ (72 * (72-22) * (72-61) 2) \u003d √ 435600 \u003d 660 cm 2. Calculations that will give the lateral surface area: 660 * 6 \u003d 3960 cm 2. It remains to add them up to find out the entire surface: 5217.47≈5217 cm 2.

Answer. Base - 726√3 cm 2, side surface - 3960 cm 2, entire area - 5217 cm 2.

We know what a cone is, let's try to find its surface area. Why is it necessary to solve such a problem? For example, you need to understand how much dough will go to make a waffle cone? Or how many bricks would it take to lay down the brick roof of a castle?

It is not easy to measure the lateral surface area of ​​a cone. But imagine the same horn wrapped in cloth. To find the area of ​​a piece of fabric, you need to cut and spread it on the table. We get a flat figure, we can find its area.

Rice. 1. Section of the cone along the generatrix

Let's do the same with the cone. Let's "cut" its lateral surface along any generatrix, for example, (see Fig. 1).

Now we “unwind” the side surface onto a plane. We get a sector. The center of this sector is the top of the cone, the radius of the sector is equal to the generatrix of the cone, and the length of its arc coincides with the circumference of the base of the cone. Such a sector is called a development of the lateral surface of the cone (see Fig. 2).

Rice. 2. Development of the side surface

Rice. 3. Angle measurement in radians

Let's try to find the area of ​​the sector according to the available data. First, let's introduce a notation: let the angle at the top of the sector be in radians (see Fig. 3).

We will often encounter the angle at the top of the sweep in tasks. In the meantime, let's try to answer the question: can't this angle turn out to be more than 360 degrees? That is, will it not turn out that the sweep will superimpose itself? Of course not. Let's prove it mathematically. Let the sweep "overlap" itself. This means that the length of the sweep arc is greater than the circumference of the radius . But, as already mentioned, the length of the sweep arc is the circumference of the radius. And the radius of the base of the cone, of course, is less than the generatrix, for example, because the leg of a right triangle is less than the hypotenuse

Then let's remember two formulas from the course of planimetry: arc length. Sector area: .

In our case, the role is played by the generatrix , and the length of the arc is equal to the circumference of the base of the cone, that is. We have:

Finally we get:

Along with the lateral surface area, the total surface area can also be found. To do this, add the base area to the lateral surface area. But the base is a circle of radius , whose area, according to the formula, is .

Finally we have: , where is the radius of the base of the cylinder, is the generatrix.

Let's solve a couple of problems on the given formulas.

Rice. 4. Desired angle

Example 1. The development of the lateral surface of the cone is a sector with an angle at the apex. Find this angle if the height of the cone is 4 cm and the radius of the base is 3 cm (see Fig. 4).

Rice. 5. Right triangle forming a cone

By the first action, according to the Pythagorean theorem, we find the generatrix: 5 cm (see Fig. 5). Further, we know that .

Example 2. The area of ​​the axial section of the cone is , the height is . Find the total surface area (see Fig. 6).

Your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please read our privacy policy and let us know if you have any questions.

Collection and use of personal information

Personal information refers to data that can be used to identify or contact a specific person.

You may be asked to provide your personal information at any time when you contact us.

The following are some examples of the types of personal information we may collect and how we may use such information.

What personal information we collect:

• When you submit an application on the site, we may collect various information, including your name, phone number, email address, etc.

How we use your personal information:

• The personal information we collect allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
• From time to time, we may use your personal information to send you important notices and messages.
• We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
• If you enter a prize draw, contest or similar incentive, we may use the information you provide to administer such programs.

Disclosure to third parties

We do not disclose information received from you to third parties.

Exceptions:

• In the event that it is necessary - in accordance with the law, judicial order, in legal proceedings, and / or based on public requests or requests from state bodies in the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public interest purposes.
• In the event of a reorganization, merger or sale, we may transfer the personal information we collect to the relevant third party successor.

Protection of personal information

We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as from unauthorized access, disclosure, alteration and destruction.

Maintaining your privacy at the company level

To ensure that your personal information is secure, we communicate privacy and security practices to our employees and strictly enforce privacy practices.