When preparing for the exam in mathematics, students have to systematize their knowledge of algebra and geometry. I would like to combine all 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 is clear with the side faces, 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 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 incorrect one. In the USE tasks of interest to schoolchildren, there are only tasks with the correct figures at the base. Therefore, we will only talk about them.

right triangle

That is equilateral. One in which all sides are equal and denoted by the letter "a". In this case, the area of ​​\u200b\u200bthe 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 a polygon has the same designation. For the number of corners, the Latin letter n is used.

S = (n * a 2) / (4 * tg (180º/n)).

How to proceed when calculating the lateral and total surface area?

Since the base is a regular figure, 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 the lateral area of ​​\u200b\u200bthe 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.

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". General formula for the lateral surface area looks like this:

S \u003d 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 vertex (α) are given. Then it is supposed to use such a formula to calculate the lateral area of ​​\u200b\u200bthe pyramid:

S = n/2 * in 2 sin α .

Task #1

Condition. Find the total area of ​​the pyramid if its base lies with 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 \u003d 3 * 4 \u003d 12 cm. Since the apothem is known, you can immediately calculate the area of ​​\u200b\u200bthe entire lateral surface: ½ * 12 * √3 = 6√3 cm 2.

For a triangle at the base, the following area value will be obtained: (4 2 * √3) / 4 \u003d 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 cm2.

Task #2

Condition. There is a regular quadrangular pyramid. The length of the side of the base is 7 mm, the side edge is 16 mm. You need to know its surface area.

Solution. Since the polyhedron is quadrangular and regular, then its base is a square. Having learned the areas of the base and side faces, it will be possible to calculate the area of ​​\u200b\u200bthe 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 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 \u003d 267.576 mm 2.

Answer. The desired value is 267.576 mm 2.

Task #3

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

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

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 right triangle) is equal to √(3 2 + 4 2) = 5 (cm).

Now you can calculate the desired value: ½ * (4 * 6) * 5 + 6 2 \u003d 96 (cm 2).

Answer. 96 cm2.

Task #4

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

First of all, the area of ​​\u200b\u200bthe base is calculated using the above formula: (6 * 22 2) / (4 * tg (180º / 6)) \u003d 726 / (tg30º) \u003d 726√3 cm 2.

Now you need to find out the semi-perimeter of an isosceles triangle, which is a lateral face. (22 + 61 * 2): 2 = 72 cm. It remains to calculate the area of ​​\u200b\u200beach such triangle using the Heron formula, and then multiply it by six and add it to the one that turned out for the base.

Calculations using the Heron formula: √ (72 * (72-22) * (72-61) 2) \u003d √ 435600 \u003d 660 cm 2. Calculations that will give the lateral surface area: 660 * 6 \u003d 3960 cm 2. It remains to add them up to find out the entire surface: 5217.47≈5217 cm 2.

Answer. Base - 726√3 cm 2, side surface - 3960 cm 2, entire area - 5217 cm 2.

triangular pyramid A polyhedron is called a polyhedron whose base is a regular triangle.

In such a pyramid, the faces 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. Find the lateral surface area correct pyramid you can use the formula. And you can make the calculation several times faster. To do this, 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. Consider an example of calculating the area of ​​a triangular pyramid.

Task: Let the correct pyramid be given. The side of the triangle lying 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 necessary elements, find the perimeter. Remember that in a regular triangle, all sides are equal, and, therefore, the perimeter is calculated by the formula:

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. For this, the formula is used:

The formula for the area of ​​\u200b\u200bthe base of a triangular pyramid may be different. It is allowed to use any calculation of parameters for a given figure, but most often this is not required. Consider an example of calculating the area of ​​the base of a triangular pyramid.

Task: In a regular pyramid, the side of the triangle lying at the base is a = 6 cm. Calculate the area of ​​​​the base.
To calculate, we only need the length of the side of a regular triangle located at the base of the pyramid. Substitute the data in the formula:

Quite often it is required to find the total area of ​​a polyhedron. To do this, you need to add the area of ​​\u200b\u200bthe side surface and the base.

Consider an example of calculating the area of ​​a triangular pyramid.

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

We substitute the data in the formula:
Now find the area of ​​the base:
Knowing the area of ​​​​the base and lateral surface, we find the total area of ​​\u200b\u200bthe pyramid:

When calculating the area of ​​\u200b\u200ba regular pyramid, one should not forget that the base is a regular triangle and many elements of this polyhedron are equal to each other.

Pyramid, at the base of which lies regular hexagon, and the sides are formed regular triangles, is called hexagonal.

This polyhedron has many properties:

• All sides and angles of the base are equal to each other;
• All edges and dihedral coal pyramids are also equal to each other;
• The triangles forming the sides are the same, respectively, they have the same area, sides and heights.

To calculate the area of ​​a regular hexagonal pyramid, the standard formula for the lateral surface area of ​​a hexagonal pyramid is used:

where P is the perimeter of the base, a is the length of the apothem of the pyramid. In most cases, you can calculate the side area using this formula, but sometimes you can use another method. Since the side faces of the pyramid are formed by equal triangles, you can find the area of ​​​​one triangle, and then multiply it by the number of sides. There are 6 of them in a hexagonal pyramid. But this method can also be used in the calculation. Let's consider an example of calculating the lateral surface area of ​​a hexagonal pyramid.

Let a regular hexagonal pyramid be given, in which the apothem is a = 7 cm, the side of the base is b = 3 cm. Calculate the area of ​​the lateral surface of the polyhedron.
First, find the perimeter of the base. Since the pyramid is regular, it has a regular hexagon at its base. So, all its sides are equal, and the perimeter is calculated by the formula:
We substitute the data in the formula:
Now we can easily find the lateral surface area by substituting the found value into the main formula:

Also an important point is the search for the area of ​​\u200b\u200bthe base. The formula for the area of ​​the base of a hexagonal pyramid is derived from the properties of a regular hexagon:

Let's consider an example of calculating the area of ​​the base of a hexagonal pyramid, taking the conditions from the previous example as a basis. From them we know that the side of the base is b = 3 cm. Let's substitute the data into the formula:

The formula for the area of ​​a hexagonal pyramid is the sum of the area of ​​the base and the side scan:

Consider an example of calculating the area of ​​a hexagonal pyramid.

Let a pyramid be given, at the base of which lies a regular hexagon with side b = 4 cm. The apothem of a given polyhedron is a = 6 cm. Find the total area.
We know that the total area consists of the areas of the base and the side sweep. So let's find them first. Calculate the perimeter:

Now find the lateral surface area:

Next, we calculate the area of ​​\u200b\u200bthe base in which the regular hexagon lies:

Now we can add up the results: