Polyhedra: how to find the lateral surface area of ​​a pyramid. Find the surface area of ​​a regular triangular pyramid

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is a figure whose base is an arbitrary polygon, and the side faces are represented by triangles. Their vertices lie at the same 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.
The right pyramid called a pyramid in which the sides of the base, angles, and edges are equal. Also in such a pyramid the area of ​​the 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, you need to find the area of ​​each individual triangle and add them together. If the pyramid is truncated, then its faces are represented by trapezoids. There is another formula for a regular pyramid. In it, the lateral surface area is calculated through the semi-perimeter of the base and the length of the apothem:

Let's 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, apothem a= 8 cm. Find the area of ​​the lateral surface.

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

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

In order to find the total area of ​​a polyhedron, you will 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, area of ​​a parallelogram etc.

Consider an example of calculating the area of ​​the base of a pyramid given by our conditions. Since the pyramid is regular, there is a square at its base.
Square area calculated by the formula: ,
where a is the side of the square. For us it is 6 cm. This means the area of ​​the base of the pyramid is:

Now all that remains is to find the total area of ​​the polyhedron. The formula for the area of ​​a pyramid consists of the sum of the area of ​​its base and the 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 us be given a regular pyramid, at the base of which lies a regular n-gon with side equal to a. Let h be the height of the side face, also called apothem pyramids. The area of ​​one side face is equal to 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 in the form:

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

Concerning total surface area, then we simply add the area of ​​the base to the side one.

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 a 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 discarded pyramid is homothetic to the large pyramid with the center of homothety at the apex. The similarity coefficient is equal to the ratio of heights: k=h 2 /h 1, or side edges, or other corresponding linear dimensions of both pyramids. We know that the areas of similar figures are related like squares of linear dimensions; so the areas of the bases of both pyramids (i.e. the area of ​​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 lateral surfaces of the pyramids are in the same relation. A similar rule exists for volumes.

Volumes of similar bodies are related like cubes of their linear dimensions; for example, the volumes of pyramids are related as the product of their heights and the area of ​​the bases, from which our rule is immediately obtained. It is of a completely general nature and directly follows from the fact that volume always has a dimension of the third power of length. Using this rule, we derive a formula expressing the volume of a truncated pyramid through the height and area 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 a full pyramid, then the coefficient of similarity between the full pyramid and the small pyramid can easily be found as the root of the ratio S 2 /S 1 . The height of a truncated pyramid is expressed as h = h 1 - h 2 = h 1 (1 - k). Now we have for the volume of a truncated pyramid (V 1 and V 2 denote the volumes of the full and small pyramids)

formula for the volume of a truncated pyramid

Let us 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 reason in exactly the same way as when deriving the formula for volume. We supplement the pyramid with the upper part, we have P 2 = kP 1, S 2 = 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 areas of the lateral surfaces of the entire resulting pyramid and its the upper part accordingly. For the lateral surface we find (a 1 and a 2 are apothems of the pyramids, a = a 1 - a 2 = a 1 (1-k))

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

Instructions

First of all, it is worth understanding that the lateral surface of the pyramid is represented by several triangles, the areas of which can be found using a variety of formulas, depending on the known data:

S = (a*h)/2, where h is the height lowered to side a;

S = a*b*sinβ, where a, b are the sides of the triangle, and β is the angle between these sides;

S = (r*(a + b + c))/2, where a, b, c are the sides of the triangle, and r is the radius of the circle inscribed in this triangle;

S = (a*b*c)/4*R, where R is the radius of the triangle circumscribed around the circle;

S = (a*b)/2 = r² + 2*r*R (if the triangle is right-angled);

S = S = (a²*√3)/4 (if the triangle is equilateral).

In fact, these are only the most basic known formulas for finding the area of ​​a triangle.

Having calculated the areas of all triangles that are the faces of the pyramid using the above formulas, you can begin to calculate the area of ​​this pyramid. This is done extremely simply: you need to add up the areas of all the triangles that form the side surface of the pyramid. This can be expressed by the formula:

Sp = ΣSi, where Sp is the area of ​​the lateral surface, Si is the area of ​​the i-th triangle, which is part of its lateral surface.

For greater clarity, we can consider a small example: given a regular pyramid, the side faces of which are formed by equilateral triangles, and at its base lies a square. The length of the edge of this pyramid is 17 cm. It is required to find the area of ​​the lateral surface of this pyramid.

Solution: the length of the edge of this pyramid is known, it is known that its faces are equilateral triangles. Thus, we can say that all sides of all triangles on the lateral surface are equal to 17 cm. Therefore, in order to calculate the area of ​​​​any of these triangles, you will need to apply the formula:

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

It is known that at the base of the pyramid lies a square. Thus, it is clear that there are four given equilateral triangles. Then the area of ​​the lateral surface of the pyramid is calculated as follows:

125.137 cm² * 4 = 500.548 cm²

Answer: The lateral surface area of ​​the pyramid is 500.548 cm²

First, let's calculate the area of ​​the lateral surface of the pyramid. The lateral surface is the sum of the areas of all lateral faces. If you are dealing with a regular pyramid (that is, one that has a regular polygon at its base, and the vertex is projected into the center of this polygon), then to calculate the entire lateral surface it is enough to multiply the perimeter of the base (that is, the sum of the lengths of all sides of the polygon lying at the base pyramid) by the height of the side face (otherwise called the apothem) and divide the resulting value by 2: Sb = 1/2P*h, where Sb is the area of ​​the side surface, P is the perimeter of the base, h is the height of the side face (apothem).

If you have an arbitrary pyramid in front of you, you will have to separately calculate the areas of all the faces and then add them up. Since the side faces of the pyramid are triangles, use the formula for the area of ​​a triangle: S=1/2b*h, where b is the base of the triangle, and h is the height. When the areas of all the faces have been calculated, all that remains is to add them up to get the area of ​​the lateral surface of the pyramid.

Then you need to calculate the area of ​​the base of the pyramid. The choice of formula for calculation depends on which polygon lies at the base of the pyramid: regular (that is, one with all sides of the same length) or irregular. The area of ​​a regular polygon can be calculated by multiplying the perimeter by the radius of the inscribed circle in the polygon and dividing the resulting value by 2: Sn = 1/2P*r, where Sn is the area of ​​the polygon, P is the perimeter, and r is the radius of the inscribed circle in the polygon .

A truncated pyramid is a polyhedron that is formed by a pyramid and its cross section parallel to the base. Finding the lateral surface area of ​​the pyramid is not difficult at all. Its very simple: the area is equal to the product of half the sum of the bases by . Let's consider an example of calculating the lateral surface area. Suppose we are given a regular pyramid. The lengths of the base are b = 5 cm, c = 3 cm. Apothem a = 4 cm. To find the area of ​​the lateral surface of the pyramid, you must first find the perimeter of the bases. In a large base it will be equal to p1=4b=4*5=20 cm. In a smaller base the formula will be as follows: p2=4c=4*3=12 cm. Therefore, the area will be equal to: s=1/2(20+12 )*4=32/2*4=64 cm.

Triangular pyramid is a polyhedron whose base is a regular triangle.

In such a pyramid, the edges of the base and the edges of the sides are equal to each other. Accordingly, the area of ​​the side faces is found from the sum of the areas of three identical triangles. You can find the lateral surface area of ​​a regular pyramid using the formula. And you can make the calculation several times faster. To do this, you need to apply the formula for the area of ​​the lateral surface of a triangular pyramid:

where p is the perimeter of the base, all sides of which are equal to b, a is the apothem lowered from the top to this base. Let's consider an example of calculating the area of ​​a triangular pyramid.

Problem: Let a regular pyramid be given. The side of the triangle at the base is b = 4 cm. The apothem of the pyramid is a = 7 cm. Find the area of ​​the lateral surface of the pyramid.
Since, according to the conditions of the problem, we know the lengths of all the necessary elements, we will find the perimeter. We remember that in a regular triangle all sides are equal, and, therefore, the perimeter is calculated by the formula:

Let's substitute the data and find the value:

Now, knowing the perimeter, we can calculate the lateral surface area:

To apply the formula for the area of ​​a triangular pyramid to calculate the full value, you need to find the area of ​​the base of the polyhedron. To do this, use the formula:

The formula for the area of ​​the base of a triangular pyramid may be different. It is possible to use any calculation of parameters for a given figure, but most often this is not required. Let's consider an example of calculating the area of ​​the base of a triangular pyramid.

Problem: In a regular pyramid, the side of the triangle at the base is a = 6 cm. Calculate the area of ​​the base.
To calculate, we only need the length of the side of the regular triangle located at the base of the pyramid. Let's substitute the data into the formula:

Quite often you need to find the total area of ​​a polyhedron. To do this, you will need to add up the area of ​​the side surface and the base.

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

Problem: Let a regular triangular pyramid be given. The base side is b = 4 cm, the apothem is a = 6 cm. Find the total area of ​​the pyramid.
First, let's find the area of ​​the lateral surface using the already known formula. Let's calculate the perimeter:

Substitute the data into the formula:
Now let's find the area of ​​the base:
Knowing the area of ​​the base and lateral surface, we find the total area of ​​the pyramid:

When calculating the area of ​​a regular pyramid, you should not forget that the base is a regular triangle and many elements of this polyhedron are equal to each other.