Area of ​​a triangular pyramid. Lateral surface area of ​​different pyramids Quadrilateral pyramid base area formula

What figure do we call a pyramid? Firstly, it is a polyhedron. Secondly, at the base of this polyhedron there is an arbitrary polygon, and the sides of the pyramid (side faces) necessarily have the shape of triangles converging at one common vertex. Now, having understood the term, let’s find out how to find the surface area of ​​the pyramid.

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

Calculating the area of ​​the base of a pyramid

The choice of calculation formula depends on the shape of the polygon underlying our pyramid. It can be regular, that is, with sides of the same length, or irregular. Let's consider both options.

The base is a regular polygon

From the school course we know:

• 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 perimeter of this polygon (P) by the radius of the circle inscribed in it (r), and then divide the result by two: Sn=1/2P*r .

At the base is 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 lowered to this base), add up all the results.

Lateral surface area of ​​the pyramid

Now let’s calculate the area of ​​the lateral surface of the pyramid, i.e. the sum of the areas of all its lateral sides. There are also 2 options 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 a pyramid, by definition, can only be triangles, the calculation is carried out using the above-mentioned formula: S=1/2a*h.
2. Let our pyramid be correct, i.e. at its base lies a regular polygon, and the projection of the top of the pyramid is at its center. Then, to calculate the area of ​​the lateral surface (Sb), it is enough to find half the product of the perimeter of the base polygon (P) and 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

For example, let's algebraically calculate 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 So=1/2a*h we find the area of ​​the base. We use the same formula to find the area of ​​each face of the pyramid, which also has a triangular shape, and we get 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. By adding up the areas of the sides and base, we obtain the total surface area of ​​the desired pyramid: Sp= So+ Sb.

Surface area of ​​a quadrangular pyramid

The area of ​​the lateral surface 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 quadrilateral - regular or irregular. The total surface area of ​​the pyramid is again obtained by adding the area of ​​the base and the total surface area of ​​the given pyramid.

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 come in 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 a truncated pyramid.
• Section- a 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 is valid only in relation to 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.

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. It is very simple: the area is equal to the product of half the sum of the bases by the apothem. Let's consider an example of calculating the lateral surface area of ​​a truncated pyramid. Suppose we are given a regular quadrangular 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.

A pyramid is a polyhedron, one of whose faces (base) is an arbitrary polygon, and the remaining faces (sides) are triangles having a common vertex. According to the number of angles, the base of the pyramid is triangular (tetrahedron), quadrangular, and so on.

A pyramid is a polyhedron with a base in the form of a polygon, and the remaining faces are triangles with a common vertex. An apothem is the height of the side face of a regular pyramid, which is drawn from its vertex.

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

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

Solution.

At the base of a regular triangular pyramid lies an equilateral triangle.
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 area of ​​the base 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. According to the problem, the angle OKM is 45 degrees.
Thus:
OK / MK = cos 45
Let's take advantage 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 = √3/6a
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.
Sside = 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

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

Solution.

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

We find the radius of a circle circumscribed around an equilateral triangle 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 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 lateral surface area 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, there is an equilateral triangle 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. According to the problem, the angle OKM is 45 degrees.
Thus:
OK / MK = cos 45
Let's take advantage