Lateral surface area of ​​a regular quadrangular pyramid: formulas and example problems. How to find the lateral surface area of ​​a pyramid The lateral surface area of ​​a pyramid is equal to the sum

is a multifaceted figure, the base of which is a polygon, and the remaining faces are represented by triangles with a common vertex.

If the base is a square, 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 area apothem– the height of the side face, lowered from its top.
The formula for the area of ​​the lateral surface of a pyramid is the sum of the areas of its lateral faces, which are equal to each other. However, this method of calculation 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 edges of the base are equal, the perimeter of the pentagon will be equal to:
Now you can find the lateral area of ​​the pyramid:

Area of ​​a regular triangular pyramid

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

Given a pyramid with apothem a = 4 cm and base face b = 2 cm. Find the area of ​​the lateral surface of the pyramid.
First, find the area of ​​one of the side faces. IN in this case She 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:

Area of ​​a truncated pyramid

Truncated A pyramid is a polyhedron that is formed by a pyramid and its cross 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 and the apothem:

Definition of a pyramid

Pyramid is a polyhedron, the base of which is a polygon, and its faces are triangles.

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It is worth dwelling on the definition of some components of the pyramid.

She, like other polyhedra, has ribs. They converge to one point called top pyramids. It can be based on an arbitrary polygon. Edge called geometric figure, formed by one of the sides of the base and two nearest ribs. In our case it is a triangle. Height pyramid is the distance from the plane in which its base lies to the top of the polyhedron. For a regular pyramid, there is also a concept apothems- this is a perpendicular descended from the top of the pyramid to its base.

Types of pyramids

There are 3 types of pyramids:

1. Rectangular- one in which any edge forms a right angle with the base.
2. Correct- its base is a regular geometric figure, and the vertex of the polygon itself is a projection of the center of the base.
3. Tetrahedron- a pyramid made up of triangles. Moreover, each of them can be taken as a basis.

Formula for surface area of ​​a pyramid

To find the total surface area of ​​the pyramid, you need to add the area of ​​the lateral surface and the area of ​​the base.

The simplest case is the case of a regular pyramid, so we will deal with it. Let us calculate the total surface area of ​​such a pyramid. The lateral surface area is:

S side = 1 2 ⋅ l ⋅ p S_(\text(side))=\frac(1)(2)\cdot l\cdot pS side​ = 2 1 ​ ⋅ l ⋅p

Ll l- apothem of the pyramid;
p p p- the perimeter of the base of the pyramid.

Total surface area of ​​the pyramid:

S = S side + S main S=S_(\text(side))+S_(\text(main))S=S side​ + S basic​

S side S_(\text(side)) S side​ - area of ​​the lateral surface of the pyramid;
S main S_(\text(basic)) S basic​ - area of ​​the base of the pyramid.

An example of solving a problem.

Find the total area of ​​a triangular pyramid if its apothem is 8 (cm), and at the base there is an equilateral triangle with side 3 (cm)

Solution

L = 8 l=8 l =8
a = 3 a=3 a =3

Let's find the perimeter of the base. Since the base is an equilateral triangle with side a a a, then its perimeter p p p(sum of all its sides):

P = a + a + a = 3 ⋅ a = 3 ⋅ 3 = 9 p=a+a+a=3\cdot a=3\cdot 3=9p =a +a +a =3 ⋅ a =3 ⋅ 3 = 9

Then the lateral area of ​​the pyramid is:

S side = 1 2 ⋅ l ⋅ p = 1 2 ⋅ 8 ⋅ 9 = 36 S_(\text(side))=\frac(1)(2)\cdot l\cdot p=\frac(1)(2) \cdot 8\cdot 9=36S side​ = 2 1 ​ ⋅ l ⋅p =2 1 ​ ⋅ 8 ⋅ 9 = 3 6 (see sq.)

Now let's find the area of ​​the base of the pyramid, that is, the area of ​​the triangle. In our case, the triangle is equilateral and its area can be calculated using the formula:

S main = 3 ⋅ a 2 4 S_(\text(basic))=\frac(\sqrt(3)\cdot a^2)(4)S basic​ = 4 3 ​ ⋅ a 2 ​

A a a- side of the triangle.

We get:

S main = 3 ⋅ a 2 4 = 3 ⋅ 3 2 4 ≈ 3.9 S_(\text(basic))=\frac(\sqrt(3)\cdot a^2)(4)=\frac(\sqrt(3 )\cdot 3^2)(4)\approx3.9S basic​ = 4 3 ​ ⋅ a 2 ​ = 4 3 ​ ⋅ 3 2 ​ ≈ 3 . 9 (see sq.)

Total area:

S = S side + S main ≈ 36 + 3.9 = 39.9 S=S_(\text(side))+S_(\text(main))\approx36+3.9=39.9S=S side​ + S basic​ ≈ 3 6 + 3 . 9 = 3 9 . 9 (see sq.)

Answer: 39.9 cm sq.

Another example, a little more complicated.

The base of the pyramid is a square with an area of ​​36 (cm2). The apothem of a polyhedron is 3 times the side of the base a a a. Find the total surface area of ​​this figure.

Solution

S quad = 36 S_(\text(quad))=36S quad​ = 3 6
l = 3 ⋅ a l=3\cdot a l =3 ⋅ a

Let's find the side of the base, that is, the side of the square. Its area and side length are related:

S quad = a 2 S_(\text(quad))=a^2S quad​ = a 2
36 = a 2 36=a^2 3 6 = a 2
a = 6 a=6 a =6

Let's find the perimeter of the base of the pyramid (that is, the perimeter of the square):

P = a + a + a + a = 4 ⋅ a = 4 ⋅ 6 = 24 p=a+a+a+a=4\cdot a=4\cdot 6=24p =a +a +a +a =4 ⋅ a =4 ⋅ 6 = 2 4

Let's find the length of the apothem:

L = 3 ⋅ a = 3 ⋅ 6 = 18 l=3\cdot a=3\cdot 6=18l =3 ⋅ a =3 ⋅ 6 = 1 8

In our case:

S quad = S main S_(\text(quad))=S_(\text(basic))S quad​ = S basic​

All that remains is to find the area of ​​the lateral surface. According to the formula:

S side = 1 2 ⋅ l ⋅ p = 1 2 ⋅ 18 ⋅ 24 = 216 S_(\text(side))=\frac(1)(2)\cdot l\cdot p=\frac(1)(2) \cdot 18\cdot 24=216S side​ = 2 1 ​ ⋅ l ⋅p =2 1 ​ ⋅ 1 8 ⋅ 2 4 = 2 1 6 (see sq.)

Total area:

$S=S^{\text{side + Sbasic = 2 1 6 + 3 6 = 2 5 2 (see sq.)}}$

Answer: 252 cm sq.

In a regular triangular pyramid SABC R- middle of the rib AB, S- top.
It is known that SR = 6, and the lateral surface area is equal to 36 .
Find the length of the segment B.C..

Let's make a drawing. In a regular pyramid, the side faces are isosceles triangles.

Line segment S.R.- the median lowered to the base, and therefore the height of the side face.

The lateral surface area of ​​a regular triangular pyramid is equal to the sum of the areas
three equal side faces S side = 3 S ABS. From here S ABS = 36: 3 = 12- area of ​​the face.

The area of ​​a triangle is equal to half the product of its base and height
S ABS = 0.5 AB SR. Knowing the area and height, we find the side of the base AB = BC.
12 = 0.5 AB 6
12 = 3 AB
AB = 4

Answer: 4

You can approach the problem from the other end. Let the base side AB = BC = a.
Then the area of ​​the face S ABS = 0.5 AB SR = 0.5 a 6 = 3a.

The area of ​​each of the three faces is equal to 3a, the area of ​​the three faces is equal 9a.
According to the conditions of the problem, the area of ​​the lateral surface of the pyramid is 36.
S side = 9a = 36.
From here a = 4.

Before studying questions about this geometric figure and its properties, you should understand some terms. When a person hears about a pyramid, he imagines huge buildings in Egypt. This is what the simplest ones look like. But they happen different types and shapes, which means the calculation formula for geometric shapes will be different.

Types of figure

Pyramid - geometric figure, denoting and representing several faces. In essence, this is the same polyhedron, at the base of which lies a polygon, and on the sides there are triangles that connect at one point - the vertex. The figure comes in two main types:

• correct;
• truncated.

In the first case, the base is a regular polygon. Here all lateral surfaces are equal between themselves and the figure itself will please the eye of a perfectionist.

In the second case, there are two bases - a large one at the very bottom and a small one between the top, repeating the shape of the main one. In other words, a truncated pyramid is a polyhedron with a cross section formed parallel to the base.

Terms and symbols

Key terms:

• Regular (equilateral) triangle- a figure with three equal angles and equal sides. In this case, all angles are 60 degrees. The figure is the simplest of regular polyhedra. If this figure lies at the base, then such a polyhedron will be called regular triangular. If the base is a square, the pyramid will be called a regular quadrangular pyramid.
• Vertex– the highest point where the edges meet. The height of the apex is formed by a straight line extending from the apex to the base of the pyramid.
• Edge– one of the planes of the polygon. It can be in the form of a triangle in the case of a triangular pyramid or in the form of a trapezoid for truncated pyramid.
• Section – flat figure, formed as a result of dissection. It should not be confused with a section, since a section also shows what is behind the section.
• Apothem- a segment drawn from the top of the pyramid to its base. It is also the height of the face where the second height point is located. This definition valid only for a regular polyhedron. For example, if this is not a truncated pyramid, then the face will be a triangle. In this case, the height of this triangle will become the apothem.

Area formulas

Find the lateral surface area of ​​the pyramid any type can be done in several ways. If the figure is not symmetrical and is a polygon with different sides, then in this case it is easier to calculate the total surface area through the totality of all surfaces. In other words, you need to calculate the area of ​​each face and add them together.

Depending on what parameters are known, formulas for calculating a square, trapezoid, arbitrary quadrilateral, etc. may be required. The formulas themselves in different cases will also have differences.

In the case of a regular figure, finding the area is much easier. It is enough to know just a few key parameters. In most cases, calculations are required specifically for such figures. Therefore, the corresponding formulas will be given below. Otherwise, you would have to write everything out over several pages, which would only confuse and confuse you.

Basic formula for calculation The lateral surface area of ​​a regular pyramid will have the following form:

S=½ Pa (P is the perimeter of the base, and is the apothem)

Let's look at one example. The polyhedron has a base with segments A1, A2, A3, A4, A5, and all of them are equal to 10 cm. Let the apothem be equal to 5 cm. First you need to find the perimeter. Since all five faces of the base are the same, you can find it like this: P = 5 * 10 = 50 cm. Next, we apply the basic formula: S = ½ * 50 * 5 = 125 cm squared.

Lateral surface area of ​​a regular triangular pyramid easiest to calculate. The formula looks like this:

S =½* ab *3, where a is the apothem, b is the face of the base. The factor of three here means the number of faces of the base, and the first part is the area of ​​the side surface. Let's look at an example. Given a figure with an apothem of 5 cm and a base edge of 8 cm. We calculate: S = 1/2*5*8*3=60 cm squared.

Lateral surface area of ​​a truncated pyramid It's a little more difficult to calculate. The formula looks like this: S =1/2*(p_01+ p_02)*a, where p_01 and p_02 are the perimeters of the bases, and is the apothem. Let's look at an example. Let’s say that for a quadrangular figure the dimensions of the sides of the bases are 3 and 6 cm, and the apothem is 4 cm.

Here, first you need to find the perimeters of the bases: р_01 =3*4=12 cm; р_02=6*4=24 cm. It remains to substitute the values ​​into the main formula and we get: S =1/2*(12+24)*4=0.5*36*4=72 cm squared.

Thus, you can find the lateral surface area of ​​a regular pyramid of any complexity. You should be careful and not confuse these calculations with the total area of ​​the entire polyhedron. And if you still need to do this, just calculate the area of ​​the largest base of the polyhedron and add it to the area of ​​the lateral surface of the polyhedron.

Video

This video will help you consolidate information on how to find the lateral surface area of ​​different pyramids.

Pyramid- one of the varieties of a polyhedron formed from polygons and triangles that lie at the base and are its faces.

Moreover, at the top of the pyramid (i.e. at one point) all the faces are united.

In order to calculate the area of ​​a pyramid, it is worth determining that its lateral surface consists of several triangles. And we can easily find their areas using

various formulas. Depending on what data we know about the triangles, we look for their area.

We list some formulas that can be used to find the area of ​​triangles:

1. S = (a*h)/2 . In this case, we know the height of the triangle h , which is lowered to the side a .
2. S = a*b*sinβ . Here are the sides of the triangle a , b , and the angle between them is β .
3. S = (r*(a + b + c))/2 . Here are the sides of the triangle a, b, c . The radius of a circle that is inscribed in a triangle is r .
4. S = (a*b*c)/4*R . The radius of a circumscribed circle around a triangle is R .
5. S = (a*b)/2 = r² + 2*r*R . This formula should only be applied when the triangle is right-angled.
6. S = (a²*√3)/4 . We apply this formula to an equilateral triangle.

Only after we calculate the areas of all the triangles that are the faces of our pyramid can we calculate the area of ​​its lateral surface. To do this, we will use the above formulas.

In order to calculate the area of ​​the lateral surface of a pyramid, no difficulties arise: you need to find out the sum of the areas of all triangles. Let's express this with the formula:

Sp = ΣSi

Here Si is the area of ​​the first triangle, and S P - area of ​​the lateral surface of the pyramid.

Let's look at an example. Given a regular pyramid, its lateral faces are formed by several equilateral triangles,

« Geometry is the most powerful tool for sharpening our mental abilities».

Galileo Galilei.

and the square is the base of the pyramid. Moreover, the edge of the pyramid has a length of 17 cm. Let's find the area lateral surface of this pyramid.

We reason like this: we know that the faces of the pyramid are triangles, they are equilateral. We also know what the edge length of this pyramid is. It follows that all triangles have equal sides and their length is 17 cm.

To calculate the area of ​​each of these triangles, you can use the following formula:

S = (17²*√3)/4 = (289*1.732)/4 = 125.137 cm²

So, since we know that the square lies at the base of the pyramid, it turns out that we have four equilateral triangles. This means that the area of ​​the lateral surface of the pyramid can be easily calculated using the following formula: 125.137 cm² * 4 = 500.548 cm²

Our answer is as follows: 500.548 cm² - this is the area of ​​the lateral surface of this pyramid.