How to find the lateral surface area of ​​a triangular pyramid. Lateral surface area of ​​different pyramids

When preparing for the exam in mathematics, students have to systematize their knowledge of algebra and geometry. I would like to combine all the 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 with the side faces is clear, 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 shape: from an arbitrary triangle to an n-gon. And this base, in addition to the difference in the number of angles, can be a correct figure or an incorrect one. In the USE tasks of interest to schoolchildren, only tasks with correct figures at the base are encountered. Therefore, we will only talk about them.

Regular triangle

That is, equilateral. The one in which all sides are equal and denoted by the letter "a". In this case, the area of ​​the 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 the polygon has the same symbol. For the number of angles, the Latin letter n is used.

S = (n * a 2) / (4 * tg (180º / n)).

What to do when calculating the lateral and total surface area?

Since the foundation lies correct figure, then 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 side area 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.

Square isosceles triangle calculated by a formula in which half of 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 the lateral surface area looks like this:

S = 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 plane angle at its apex (α) are given. Then it is supposed to use the following formula to calculate the lateral area of ​​the pyramid:

S = n / 2 * in 2 sin α .

Problem number 1

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

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

For a triangle at the base, you get the following area value: (4 2 * √3) / 4 = 4√3 cm 2.

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

Answer. 10√3 cm 2.

Problem number 2

Condition... There is a regular quadrangular pyramid. The length of the side of the base is 7 mm, the lateral rib is 16 mm. It is necessary to find out its surface area.

Solution. Since the polyhedron is quadrangular and regular, there is a square at its base. Having learned the areas of the base and side faces, it will be possible to calculate the area of ​​the 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 need to calculate the half-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 = 267.576 mm 2.

Answer... The desired value is 267.576 mm 2.

Problem number 3

Condition... Have the correct quadrangular pyramid it is necessary to calculate the area. The side of the square is known in it - 6 cm and the height - 4 cm.

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

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 (hypotenuse of a right-angled triangle) is √ (3 2 + 4 2) = 5 (cm).

Now you can calculate the required value: ½ * (4 * 6) * 5 + 6 2 = 96 (cm 2).

Answer. 96 cm 2.

Problem number 4

Condition. Dana correct side its bases are 22 mm, and the lateral 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 №2. Only there was given a pyramid with a square at the base, and now it is a hexagon.

The first step is to calculate the area of ​​the base according to the above formula: (6 * 22 2) / (4 * tg (180º / 6)) = 726 / (tg30º) = 726√3 cm 2.

Now you need to find out the semiperimeter of the isosceles triangle, which is the side face. (22 + 61 * 2): 2 = 72 cm. It remains to calculate the area of ​​each such triangle using Heron's formula, and then multiply it by six and add it to the one that turned out for the base.

Calculations using Heron's formula: √ (72 * (72-22) * (72-61) 2) = √435600 = 660 cm 2. Calculations that will give the lateral surface area: 660 * 6 = 3960 cm 2. It remains to fold them to find out the entire surface: 5217.47 ~ 5217 cm 2.

Answer. The base is 726√3 cm 2, the lateral surface is 3960 cm 2, the whole area is 5217 cm 2.

What shape do we call a pyramid? First, it is a polyhedron. Secondly, an arbitrary polygon is located at the base of this polyhedron, and the sides of the pyramid (side faces) necessarily have the shape of triangles converging at one common vertex. Now, having dealt with the term, we will find out how to find the surface area of ​​the pyramid.

It is clear that the surface area of ​​such geometric body will consist of the sum of the areas of the base and its entire lateral surface.

Calculating the area of ​​the base of the pyramid

The choice of the calculation formula depends on the shape of the polygon lying at the base of our pyramid. It can be correct, that is, with sides of the same length, or it can be incorrect. Let's consider both options.

At the base is a regular polygon

From school course known:

• the area of ​​the square will be equal to the length of its side squared;
• the area of ​​an equilateral triangle is equal to the square of its side divided by 4 and multiplied by the square root of three.

But there is also a general formula for calculating the area of ​​any regular polygon (Sn): you need to multiply the value of the perimeter of this polygon (P) by the radius of the inscribed circle (r), and then divide the result by two: Sn = 1 / 2P * r ...

At the base - an irregular polygon

The scheme for finding its area is to first divide the entire polygon into triangles, calculate the area of ​​each of them using the formula: 1 / 2a * h (where a is the base of the triangle, h is the height dropped to this base), add up all the results.

The area of ​​the lateral surface of the pyramid

Now let's calculate the area of ​​the side surface of the pyramid, i.e. the sum of the areas of all its lateral sides. 2 options are also possible here.

1. Let us have an arbitrary pyramid, i.e. one with an irregular polygon at its base. Then you should calculate the area of ​​each face separately and add the results. Since the sides of the pyramid, by definition, can only be triangles, the calculation is carried out according to the above formula: S = 1 / 2a * h.
2. Let our pyramid be correct, i.e. a regular polygon lies at its base, and the projection of the apex of the pyramid is in its center. Then, to calculate the area of ​​the lateral surface (Sb), it is enough to find half of the product of the perimeter of the base polygon (P) by the height (h) of the lateral side (the same for all faces): Sb = 1/2 P * h. The perimeter of a polygon is determined by adding the lengths of all its sides.

The total surface area of ​​a regular pyramid is found by summing the area of ​​its base with the area of ​​the entire lateral surface.

Examples of

As an example, let us calculate algebraically the surface areas of several pyramids.

Surface area of ​​a triangular pyramid

At the base of such a pyramid is a triangle. Using the formula Sо = 1 / 2a * h, we find the area of ​​the base. We apply the same formula to find the area of ​​each facet of the pyramid, which also has a triangular shape, and we obtain 3 areas: S1, S2 and S3. The area of ​​the lateral surface of the pyramid is the sum of all areas: Sb = S1 + S2 + S3. Adding the areas of the sides and base, we obtain the total surface area of ​​the required pyramid: Sп = Sо + Sb.

Surface area of ​​a quadrangular pyramid

The lateral surface area is the sum of 4 terms: Sb = S1 + S2 + S3 + S4, each of which is calculated using the formula for the area of ​​a triangle. And the area of ​​the base will have to be looked for, depending on the shape of the quadrangle - correct or incorrect. Square full surface the pyramid is again obtained by adding the area of ​​the base and full area surface of the given pyramid.

Typical geometric problems on the plane and in three-dimensional space are the problems of determining the areas of surfaces of different shapes. In this article, we present the formula for the lateral surface area of ​​a regular quadrangular pyramid.

What is a pyramid?

Here is a strict geometric definition of a pyramid. Suppose there is some polygon with n sides and n corners. Choose an arbitrary point in space, which will not be in the plane of the specified n-gon, and connect it to each vertex of the polygon. We get a figure with a certain volume, which is called an n-sided pyramid. For example, let's show in the figure below what a pentagonal pyramid looks like.

Two important elements of any pyramid are its base (n-gon) and its top. These elements are connected to each other by n triangles, which are generally not equal to each other. The perpendicular dropped from the top to the base is called the height of the figure. If it intersects the base at the geometric center (coincides with the center of mass of the polygon), then such a pyramid is called a straight line. If, in addition to this condition, the basis is regular polygon, then the whole pyramid is called correct. The figure below shows what regular pyramids look like with triangular, quadrangular, pentagonal and hexagonal bases.

Pyramid surface

Before proceeding to the question of the lateral surface area of ​​a regular quadrangular pyramid, one should dwell in more detail on the concept of the surface itself.

As mentioned above and shown in the figures, any pyramid is formed by a set of faces or sides. One side is the base and the n sides are triangles. The surface of the entire figure is the sum of the areas of each side of it.

It is convenient to study the surface using the example of unfolding a figure. A flat pattern for a regular quadrangular pyramid is shown in the figures below.

We see that its surface area is equal to the sum of four areas of identical isosceles triangles and the area of ​​a square.

The total area of ​​all triangles that form the lateral sides of the figure is usually called the lateral surface area. Next, we will show how to calculate it for a regular quadrangular pyramid.

Lateral surface area of ​​a quadrangular regular pyramid

To calculate the lateral surface area of ​​the specified shape, refer back to the above flat pattern. Suppose we know the side of a square base. Let us denote it by the symbol a. It can be seen that each of the four identical triangles has a base of length a. To calculate their total area, you need to know this value for one triangle. It is known from the geometry course that the area of ​​a triangle S t is equal to the product of the base and the height, which should be divided in half. That is:

Where h b is the height of an isosceles triangle drawn to the base a. For the pyramid, this height is apothem. Now it remains to multiply the resulting expression by 4 to obtain the area S b of the lateral surface for the pyramid under consideration:

S b = 4 * S t = 2 * h b * a.

This formula contains two parameters: apotema and a side of the base. If the latter is known in most of the conditions of the problems, then the former has to be calculated knowing other quantities. We give formulas for calculating apotema h b for two cases:

• when the length is known lateral rib;
• when the height of the pyramid is known.

If we denote the length of the side edge (side of an isosceles triangle) by the symbol L, then apotema h b is determined by the formula:

h b = √ (L 2 - a 2/4).

This expression is the result of applying the Pythagorean theorem to the triangle of the lateral surface.

If the height h of the pyramid is known, then apotema h b can be calculated as follows:

h b = √ (h 2 + a 2/4).

This expression is also not difficult to obtain if we consider the inside of the pyramid. right triangle formed by the legs h and a / 2 and the hypotenuse h b.

Let's show how to apply these formulas by solving two interesting problems.

Known surface area problem

It is known that the area of ​​the lateral surface of the quadrangular is 108 cm 2. It is necessary to calculate the value of the length of its apotema h b if the height of the pyramid is 7 cm.

Let us write the formula for the area S b of the lateral surface in terms of the height. We have:

S b = 2 * √ (h 2 + a 2/4) * a.

Here we simply substituted the corresponding apotema formula in the expression for S b. Let's square both sides of the equality:

S b 2 = 4 * a 2 * h 2 + a 4.

To find the value of a, let's change the variables:

t 2 + 4 * h 2 * t - S b 2 = 0.

Substitute now known values and we decide quadratic equation:

t 2 + 196 * t - 11664 = 0.

We wrote out only the positive root of this equation. Then the sides of the base of the pyramid will be equal to:

a = √t = √47.8355 ≈ 6.916 cm.

To get the length of the apotema, just use the formula:

h b = √ (h 2 + a 2/4) = √ (7 2 + 6.916 2/4) ≈ 7.808 cm.

The side surface of the Cheops pyramid

Let's determine the value of the lateral surface area for the largest Egyptian pyramid. It is known that at its base lies a square with a side length of 230.363 meters. The height of the building was originally 146.5 meters. Substituting these numbers into the corresponding formula for S b, we get:

S b = 2 * √ (h 2 + a 2/4) * a = 2 * √ (146.5 2 +230.363 2/4) * 230.363 ≈ 85860 m 2.

The found value is slightly larger than the area of ​​17 football fields.

Is a polyhedral figure, at the base of which is a polygon, and the rest of the faces are represented by triangles with a common vertex.

If there is a square at the base, then the pyramid is called quadrangular, if a triangle - then triangular... The height of the pyramid is drawn from its top perpendicular to the base. Also used to calculate the area apothem- the height of the side face dropped from its top.
The formula for the lateral surface area of ​​a pyramid is the sum of the areas of its lateral faces, which are equal to each other. However, this calculation method is used very rarely. Basically, the area of ​​the pyramid is calculated through the perimeter of the base and the apothem:

Let's consider an example of calculating the area of ​​the lateral surface of a pyramid.

Let a pyramid be given with base ABCDE and top F. AB = BC = CD = DE = EA = 3 cm. Apothem a = 5 cm. Find the area of ​​the lateral surface of the pyramid.
Let's find the perimeter. Since all the faces of the base are equal, the perimeter of the pentagon will be equal to:
Now you can find the side area of ​​the pyramid:

Area of ​​a regular triangular pyramid

A regular triangular pyramid consists of a base in which an equilateral triangle lies and three side faces that are equal in area.
The formula for the lateral surface area of ​​a regular triangular pyramid can be calculated different ways... You can apply the usual formula for calculating through the perimeter and apothem, or you can find the area of ​​one face and multiply it by three. Since the face of the pyramid is a triangle, we will apply the formula for the area of ​​a triangle. It will require apothem and base length. Let's consider an example of calculating the lateral surface area of ​​a regular triangular pyramid.

You are given a pyramid with an apothem of a = 4 cm and a face of the base b = 2 cm. Find the area of ​​the lateral surface of the pyramid.
First, find the area of ​​one of the side faces. In this case, it will be:
Substitute the values ​​into the formula:
Since in a regular pyramid all the sides are the same, the area of ​​the side surface of the pyramid will be equal to the sum of the areas of the three faces. Respectively:

Truncated pyramid area

Truncated A pyramid is a polyhedron that is formed by a pyramid and its section parallel to the base.
The formula for the lateral surface area of ​​a truncated pyramid is very simple. The area is equal to the product of half the sum of the perimeters of the bases by the apothem:

In this tutorial:

• Problem 1. Find the total surface area of ​​the pyramid
• Problem 2. Find the lateral surface area of ​​a regular triangular pyramid

See also related materials:
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Note ... If you need to solve a geometry problem that is not here, write about it in the forum. In tasks, instead of the "square root" symbol, the sqrt () function is used, in which sqrt is the square root symbol, and the radical expression is indicated in parentheses. For simple radical expressions, the "√" sign can be used.

Problem 1... Find the total surface area of ​​a regular pyramid

The height of the base of a regular triangular pyramid is 3 cm, and the angle between the side face and the base of the pyramid is 45 degrees.
Find the total surface area of ​​a pyramid

Solution.

An equilateral triangle lies at the base of a regular triangular pyramid.
Therefore, to solve the problem, we will use the properties of a regular triangle:

We know the height of the triangle, from where we can find its area.
h = √3 / 2 a
a = h / (√3 / 2)
a = 3 / (√3 / 2)
a = 6 / √3

Whence the base area will be equal to:
S = √3 / 4 a 2
S = √3 / 4 (6 / √3) 2
S = 3√3

In order to find the area of ​​the side face, we calculate the height KM. The OKM angle is 45 degrees according to the problem statement.
Thus:
OK / MK = cos 45
Let's use the table of values ​​of trigonometric functions and substitute the known values.

OK / MK = √2 / 2

Let's take into account that OK is equal to the radius of the inscribed circle. Then
OK = √3 / 6 a
OK = √3 / 6 * 6 / √3 = 1

Then
OK / MK = √2 / 2
1 / MK = √2 / 2
MK = 2 / √2

The area of ​​the side face is then equal to half the product of the height and the base of the triangle.
Side = 1/2 (6 / √3) (2 / √2) = 6 / √6

Thus, the total surface area of ​​the pyramid will be equal to
S = 3√3 + 3 * 6 / √6
S = 3√3 + 18 / √6

Answer: 3√3 + 18/√6

Task 2... Find the lateral surface area of ​​a regular pyramid

In a regular triangular pyramid, the height is 10 cm, and the side of the base is 16 cm ... Find the lateral surface area .

Solution.

Since the base of a regular triangular pyramid is an equilateral triangle, AO is the radius of a circle circumscribed around the base.
(This follows from)

The radius of a circle circumscribed around an equilateral triangle is found from its properties

Whence the length of the edges of a regular triangular pyramid will be equal to:
AM 2 = MO 2 + AO 2
the height of the pyramid is known by the condition (10 cm), AO = 16√3 / 3
AM 2 = 100 + 256/3
AM = √ (556/3)

Each side of the pyramid is an isosceles triangle. We find the area of ​​an isosceles triangle from the first formula presented below

S = 1/2 * 16 sqrt ((√ (556/3) + 8) (√ (556/3) - 8))
S = 8 sqrt ((556/3) - 64)
S = 8 sqrt (364/3)
S = 16 sqrt (91/3)

Since all three faces of a regular pyramid are equal, the area of ​​the lateral surface will be equal to
3S = 48 √ (91/3)

Answer: 48 √(91/3)

Problem 3. Find the total surface area of ​​a regular pyramid

The side of a regular triangular pyramid is 3 cm and the angle between the side face and the base of the pyramid is 45 degrees. Find the total surface area of ​​the pyramid.

Solution.
Since the pyramid is regular, an equilateral triangle lies at its base. Therefore, the area of ​​the base is


So = 9 * √3 / 4

In order to find the area of ​​the side face, we calculate the height KM. The OKM angle is 45 degrees according to the problem statement.
Thus:
OK / MK = cos 45
We will use