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

How to find 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 irregular one. In the Unified State Exam tasks that interest schoolchildren, there are only tasks with correct figures at the base. Therefore, we will talk only about them.

Regular triangle

That is, equilateral. The one in which all sides are equal and are designated 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 again the side:

Arbitrary regular n-gon

The side of a polygon has the same notation. 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 base is a regular figure, all 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 ​​the pyramid, you will 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 = 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 apex (α) are given. Then you need to use the following formula to calculate the lateral area of ​​the pyramid:

S = n/2 * in 2 sin α .

Task No. 1

Condition. Find the total area of ​​the pyramid if its base has 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 = 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 the triangle at the base, you get the following area value: (4 2 *√3) / 4 = 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 cm 2.

Problem No. 2

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

Solution. Since the polyhedron is quadrangular and regular, its base is a square. Once you know the area of ​​the base and side faces, you will be able to calculate the area of ​​the pyramid. The formula for the square is given above. And for 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 the following 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 = 267.576 mm 2.

Answer. The desired value is 267.576 mm 2.

Problem No. 3

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

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

We will 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 required apothem (hypotenuse of a right triangle) is equal to √(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 No. 4

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

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

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

Now you need to find out the semi-perimeter of an isosceles triangle, which is the side face. (22+61*2):2 = 72 cm. All that remains is to use Heron’s formula to calculate the area of ​​each such triangle, and then multiply it by six and add it to the one obtained 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 add them up to find out the entire surface: 5217.47≈5217 cm 2.

Answer. The base is 726√3 cm2, the side surface is 3960 cm2, the entire area is 5217 cm2.

Surface area of ​​the pyramid. In this article we will look at problems with regular pyramids. Let me remind you that a regular pyramid is a pyramid whose base is a regular polygon, the top of the pyramid is projected into the center of this polygon.

The side face of such a pyramid is an isosceles triangle.The altitude of this triangle drawn from the vertex of a regular pyramid is called apothem, SF - apothem:

In the type of problem presented below, you need to find the surface area of ​​the entire pyramid or the area of ​​its lateral surface. The blog has already discussed several problems with regular pyramids, where the question was about finding the elements (height, base edge, side edge).

Unified State Examination tasks usually examine regular triangular, quadrangular and hexagonal pyramids. I haven’t seen any problems with regular pentagonal and heptagonal pyramids.

The formula for the area of ​​the entire surface is simple - you need to find the sum of the area of ​​the base of the pyramid and the area of ​​its lateral surface:

Let's consider the tasks:

The sides of the base of a regular quadrangular pyramid are 72, the side edges are 164. Find the surface area of ​​this pyramid.

The surface area of ​​the pyramid is equal to the sum of the areas of the lateral surface and the base:

*The lateral surface consists of four triangles of equal area. The base of the pyramid is a square.

We can calculate the area of ​​the side of the pyramid using:

Thus, the surface area of ​​the pyramid is:

Answer: 28224

The sides of the base of a regular hexagonal pyramid are equal to 22, the side edges are equal to 61. Find the lateral surface area of ​​this pyramid.

The base of a regular hexagonal pyramid is a regular hexagon.

The lateral surface area of ​​this pyramid consists of six areas of equal triangles with sides 61,61 and 22:

Let's find the area of ​​the triangle using Heron's formula:

Thus, the lateral surface area is:

Answer: 3240

*In the problems presented above, the area of ​​the side face could be found using another triangle formula, but for this you need to calculate the apothem.

27155. Find the surface area of ​​a regular quadrangular pyramid whose base sides are 6 and whose height is 4.

In order to find the surface area of ​​the pyramid, we need to know the area of ​​the base and the area of ​​the lateral surface:

The area of ​​the base is 36 since it is a square with side 6.

The lateral surface consists of four faces, which are equal triangles. In order to find the area of ​​such a triangle, you need to know its base and height (apothem):

*The area of ​​a triangle is equal to half the product of the base and the height drawn to this base.

The base is known, it is equal to six. Let's find the height. Consider a right triangle (highlighted in yellow):

One leg is equal to 4, since this is the height of the pyramid, the other is equal to 3, since it is equal to half the edge of the base. We can find the hypotenuse using the Pythagorean theorem:

This means that the area of ​​the lateral surface of the pyramid is:

Thus, the surface area of ​​the entire pyramid is:

Answer: 96

27069. The sides of the base of a regular quadrangular pyramid are equal to 10, the side edges are equal to 13. Find the surface area of ​​this pyramid.

27070. The sides of the base of a regular hexagonal pyramid are equal to 10, the side edges are equal to 13. Find the lateral surface area of ​​this pyramid.

There are also formulas for the lateral surface area of ​​a regular pyramid. In a regular pyramid, the base is an orthogonal projection of the lateral surface, therefore:

P- base perimeter, l- apothem of the pyramid

*This formula is based on the formula for the area of ​​a triangle.

If you want to learn more about how these formulas are derived, don’t miss it, follow the publication of articles.That's all. Good luck to you!

Sincerely, Alexander Krutitskikh.

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The total area of ​​the lateral surface of a pyramid consists of the sum of the areas of its lateral faces.

In a quadrangular pyramid, there are two types of faces - a quadrangle at the base and triangles with a common vertex, which form the side surface.
First you need to calculate the area of ​​the side faces. To do this, you can use the formula for the area of ​​a triangle, or you can also use the formula for the surface area of ​​a quadrangular pyramid (only if the polyhedron is regular). If the pyramid is regular and the length of the edge a of the base and the apothem h drawn to it is known, then:

If, according to the conditions, the length of the edge c of a regular pyramid and the length of the side of the base a are given, then you can find the value using the following formula:

If the length of the edge at the base and the acute angle opposite it at the top are given, then the area of ​​the lateral surface can be calculated by the ratio of the square of the side a to the double cosine of half the angle α:

Let's consider an example of calculating the surface area of ​​a quadrangular pyramid through the side edge and the side of the base.

Problem: Let a regular quadrangular pyramid be given. Edge length b = 7 cm, base side length a = 4 cm. Substitute the given values ​​into the formula:

We showed calculations of the area of ​​one side face for a regular pyramid. Respectively. To find the area of ​​the entire surface, you need to multiply the result by the number of faces, that is, by 4. If the pyramid is arbitrary and its faces are not equal to each other, then the area must be calculated for each individual side. If the base is a rectangle or parallelogram, then it is worth remembering their properties. The sides of these figures are parallel in pairs, and accordingly the faces of the pyramid will also be identical in pairs.
The formula for the area of ​​the base of a quadrangular pyramid directly depends on which quadrilateral lies at the base. If the pyramid is correct, then the area of ​​the base is calculated using the formula, if the base is a rhombus, then you will need to remember how it is located. If there is a rectangle at the base, then finding its area will be quite simple. It is enough to know the lengths of the sides of the base. Let's consider an example of calculating the area of ​​the base of a quadrangular pyramid.

Problem: Let a pyramid be given, at the base of which lies a rectangle with sides a = 3 cm, b = 5 cm. An apothem is lowered from the top of the pyramid to each of the sides. h-a =4 cm, h-b =6 cm. The top of the pyramid lies on the same line as the point of intersection of the diagonals. Find the total area of ​​the pyramid.
The formula for the area of ​​a quadrangular pyramid consists of the sum of the areas of all faces and the area of ​​the base. First, let's find the area of ​​the base:


Now let's look at the sides of the pyramid. They are identical in pairs, because the height of the pyramid intersects the point of intersection of the diagonals. That is, in our pyramid there are two triangles with a base a and height h-a, as well as two triangles with a base b and height h-b. Now let's find the area of ​​the triangle using the well-known formula:


Now let's perform an example of calculating the area of ​​a quadrangular pyramid. In our pyramid with a rectangle at the base, the formula would look like this:

Students encounter the concept of a pyramid long before studying geometry. The fault lies with the famous great Egyptian wonders of the world. Therefore, when starting to study this wonderful polyhedron, most students already clearly imagine it. All the above-mentioned attractions have the correct shape. What's happened regular pyramid, and what properties it has will be discussed further.

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Definition

There are quite a lot of definitions of a pyramid. Since ancient times, it has been very popular.

For example, Euclid defined it as a bodily figure consisting of planes that, starting from one, converge at a certain point.

Heron provided a more precise formulation. He insisted that this was the figure that has a base and planes in the form of triangles, converging at one point.

Based on the modern interpretation, the pyramid is represented as a spatial polyhedron, consisting of a certain k-gon and k flat triangular figures, having one common point.

Let's look at it in more detail, what elements does it consist of:

• The k-gon is considered the basis of the figure;
• 3-gonal shapes protrude as the edges of the side part;
• the upper part from which the side elements originate is called the apex;
• all segments connecting a vertex are called edges;
• if a straight line is lowered from the vertex to the plane of the figure at an angle of 90 degrees, then its part contained in the internal space is the height of the pyramid;
• in any lateral element, a perpendicular, called an apothem, can be drawn to the side of our polyhedron.

The number of edges is calculated using the formula 2*k, where k is the number of sides of the k-gon. How many faces a polyhedron such as a pyramid has can be determined using the expression k+1.

Important! A pyramid of regular shape is a stereometric figure whose base plane is a k-gon with equal sides.

Basic properties

Correct pyramid has many properties, which are unique to her. Let's list them:

1. The basis is a figure of the correct shape.
2. The edges of the pyramid that limit the side elements have equal numerical values.
3. The side elements are isosceles triangles.
4. The base of the height of the figure falls at the center of the polygon, while it is simultaneously the central point of the inscribed and circumscribed.
5. All side ribs are inclined to the plane of the base at the same angle.
6. All side surfaces have the same angle of inclination relative to the base.

Thanks to all of the listed properties, performing element calculations is much simpler. Based on the above properties, we pay attention to two signs:

1. In the case when the polygon fits into a circle, the side faces will have equal angles with the base.
2. When describing a circle around a polygon, all edges of the pyramid emanating from the vertex will have equal lengths and equal angles with the base.

The basis is a square

Regular quadrangular pyramid - a polyhedron whose base is a square.

It has four side faces, which are isosceles in appearance.

A square is depicted on a plane, but is based on all the properties of a regular quadrilateral.

For example, if it is necessary to relate the side of a square with its diagonal, then use the following formula: the diagonal is equal to the product of the side of the square and the square root of two.

It is based on a regular triangle

A regular triangular pyramid is a polyhedron whose base is a regular 3-gon.

If the base is a regular triangle and the side edges are equal to the edges of the base, then such a figure called a tetrahedron.

All faces of a tetrahedron are equilateral 3-gons. In this case, you need to know some points and not waste time on them when calculating:

• the angle of inclination of the ribs to any base is 60 degrees;
• the size of all internal faces is also 60 degrees;
• any face can act as a base;
• , drawn inside the figure, these are equal elements.

Sections of a polyhedron

In any polyhedron there are several types of sections flat. Often in a school geometry course they work with two:

• axial;
• parallel to the basis.

An axial section is obtained by intersecting a polyhedron with a plane that passes through the vertex, side edges and axis. In this case, the axis is the height drawn from the vertex. The cutting plane is limited by the lines of intersection with all faces, resulting in a triangle.

Attention! In a regular pyramid, the axial section is an isosceles triangle.

If the cutting plane runs parallel to the base, then the result is the second option. In this case, we have a cross-sectional figure similar to the base.

For example, if there is a square at the base, then the section parallel to the base will also be a square, only of smaller dimensions.

When solving problems under this condition, they use signs and properties of similarity of figures, based on Thales' theorem. First of all, it is necessary to determine the similarity coefficient.

If the plane is drawn parallel to the base and it cuts off the upper part of the polyhedron, then a regular truncated pyramid is obtained in the lower part. Then the bases of a truncated polyhedron are said to be similar polygons. In this case, the side faces are isosceles trapezoids. The axial section is also isosceles.

In order to determine the height of a truncated polyhedron, it is necessary to draw the height in the axial section, that is, in the trapezoid.

Surface areas

The main geometric problems that have to be solved in a school geometry course are finding the surface area and volume of a pyramid.

There are two types of surface area values:

• area of ​​the side elements;
• area of ​​the entire surface.

From the name itself it is clear what we are talking about. The side surface includes only the side elements. It follows from this that to find it, you simply need to add up the areas of the lateral planes, that is, the areas of isosceles 3-gons. Let's try to derive the formula for the area of ​​the side elements:

1. The area of ​​an isosceles 3-gon is Str=1/2(aL), where a is the side of the base, L is the apothem.
2. The number of lateral planes depends on the type of k-gon at the base. For example, a regular quadrangular pyramid has four lateral planes. Therefore, it is necessary to add the areas of four figures Sside=1/2(aL)+1/2(aL)+1/2(aL)+1/2(aL)=1/2*4a*L. The expression is simplified in this way because the value is 4a = Rosn, where Rosn is the perimeter of the base. And the expression 1/2*Rosn is its semi-perimeter.
3. So, we conclude that the area of ​​the lateral elements of a regular pyramid is equal to the product of the semi-perimeter of the base and the apothem: Sside = Rosn * L.

The area of ​​the total surface of the pyramid consists of the sum of the areas of the side planes and the base: Sp.p. = Sside + Sbas.

As for the area of ​​the base, here the formula is used according to the type of polygon.

Volume of a regular pyramid equal to the product of the area of ​​the base plane and the height divided by three: V=1/3*Sbas*H, where H is the height of the polyhedron.

What is a regular pyramid in geometry

Properties of a regular quadrangular pyramid

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 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:

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: