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". The general formula for lateral surface area is:
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.
Decision. 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.
Decision. 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.
Decision. 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 (the hypotenuse of a right triangle) is √(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?
Decision. 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. 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, 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 the necessary elements, we will 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 using the already 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.
Definition. Side face- this is a triangle in which one angle lies at the top of the pyramid, and the opposite side of it coincides with the side of the base (polygon).
Definition. Side ribs are the common sides of the side faces. A pyramid has as many edges as there are corners in a polygon.
Definition. pyramid height is a perpendicular dropped from the top to the base of the pyramid.
Definition. Apothem- this is the perpendicular of the side face of the pyramid, lowered from the top of the pyramid to the side of the base.
Definition. Diagonal section- this is a section of the pyramid by a plane passing through the top of the pyramid and the diagonal of the base.
Definition. Correct pyramid- This is a pyramid in which the base is a regular polygon, and the height descends to the center of the base.
Volume and surface area of the pyramid
Formula. pyramid volume through base area and height:
pyramid properties
If all side edges are equal, then a circle can be circumscribed around the base of the pyramid, and the center of the base coincides with the center of the circle. Also, the perpendicular dropped from the top passes through the center of the base (circle).
If all side ribs are equal, then they are inclined to the base plane at the same angles.
The lateral ribs are equal when they form equal angles with the base plane, or if a circle can be described around the base of the pyramid.
If the side faces are inclined to the plane of the base at one angle, then a circle can be inscribed in the base of the pyramid, and the top of the pyramid is projected into its center.
If the side faces are inclined to the base plane at one angle, then the apothems of the side faces are equal.
Properties of a regular pyramid
1. The top of the pyramid is equidistant from all corners of the base.
2. All side edges are equal.
3. All side ribs are inclined at the same angles to the base.
4. Apothems of all side faces are equal.
5. The areas of all side faces are equal.
6. All faces have the same dihedral (flat) angles.
7. A sphere can be described around the pyramid. The center of the described sphere will be the intersection point of the perpendiculars that pass through the middle of the edges.
8. A sphere can be inscribed in a pyramid. The center of the inscribed sphere will be the intersection point of the bisectors emanating from the angle between the edge and the base.
9. If the center of the inscribed sphere coincides with the center of the circumscribed sphere, then the sum of the flat angles at the apex is equal to π or vice versa, one angle is equal to π / n, where n is the number of angles at the base of the pyramid.
The connection of the pyramid with the sphere
A sphere can be described around the pyramid when at the base of the pyramid lies a polyhedron around which a circle can be described (a necessary and sufficient condition). The center of the sphere will be the point of intersection of planes passing perpendicularly through the midpoints of the side edges of the pyramid.
A sphere can always be described around any triangular or regular pyramid.
A sphere can be inscribed in a pyramid if the bisector planes of the internal dihedral angles of the pyramid intersect at one point (a necessary and sufficient condition). This point will be the center of the sphere.
The connection of the pyramid with the cone
A cone is called inscribed in a pyramid if their vertices coincide and the base of the cone is inscribed in the base of the pyramid.
A cone can be inscribed in a pyramid if the apothems of the pyramid are equal.
A cone is said to be circumscribed around a pyramid if their vertices coincide and the base of the cone is circumscribed around the base of the pyramid.
A cone can be described around a pyramid if all side edges of the pyramid are equal to each other.
Connection of a pyramid with a cylinder
A pyramid is said to be inscribed in a cylinder if the top of the pyramid lies on one base of the cylinder, and the base of the pyramid is inscribed in another base of the cylinder.
A cylinder can be circumscribed around a pyramid if a circle can be circumscribed around the base of the pyramid.
A tetrahedron has four faces and four vertices and six edges, where any two edges have no common vertices but do not touch.
Each vertex consists of three faces and edges that form trihedral angle.
The segment connecting the vertex of the tetrahedron with the center of the opposite face is called median of the tetrahedron(GM).
Bimedian is called a segment connecting the midpoints of opposite edges that do not touch (KL).
All bimedians and medians of a tetrahedron intersect at one point (S). In this case, the bimedians are divided in half, and the medians in a ratio of 3: 1 starting from the top.
Definition. Acute Angled Pyramid is a pyramid in which the apothem is more than half the length of the side of the base.
Definition. obtuse pyramid is a pyramid in which the apothem is less than half the length of the side of the base.
Definition. regular tetrahedron A tetrahedron whose four faces are equilateral triangles. It is one of five regular polygons. In a regular tetrahedron, all dihedral angles (between faces) and trihedral angles (at a vertex) are equal.
Definition. Rectangular tetrahedron a tetrahedron is called which has a right angle between three edges at the vertex (the edges are perpendicular). Three faces form rectangular trihedral angle and the faces are right triangles, and the base is an arbitrary triangle. The apothem of any face is equal to half the side of the base on which the apothem falls.
Definition. Isohedral tetrahedron A tetrahedron is called in which the side faces are equal to each other, and the base is a regular triangle. The faces of such a tetrahedron are isosceles triangles.
Definition. Orthocentric tetrahedron a tetrahedron is called in which all the heights (perpendiculars) that are lowered from the top to the opposite face intersect at one point.
Definition. star pyramid A polyhedron whose base is a star is called.
A pyramid whose base is a regular hexagon, and the sides are formed by 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:
