How to find the perimeter of a pyramid formula. Area of a triangular pyramid. What to do when finding the area of the base of the pyramid
Definition. Side edge is a triangle, one corner of which lies at the top of the pyramid, and the opposite side coincides with the side of the base (polygon).
Definition. Side ribs are the common sides of the side faces. The pyramid has as many edges as the corners of the polygon.
Definition. Pyramid height- this is a perpendicular, lowered from the top to the base of the pyramid.
Definition. Apothem is the perpendicular to the side face of the pyramid, lowered from the top of the pyramid to the side of the base.
Definition. Diagonal section 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 is a pyramid in which the base is a regular polygon, and the height drops to the center of the base.
Volume and surface area of the pyramid
Formula. The volume of the pyramid through the base area and height:
Pyramid properties
If all side edges are equal, then a circle can be described 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 edges are equal, then they are inclined to the plane of the base at the same angles.
The side ribs are equal when they form with the plane of the base equal angles or if a circle can be described around the base of the pyramid.
If the side faces are inclined to the base plane at one angle, then a circle can be inscribed into 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 the same 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 slope at the same angle to the base.
4. The apothems of all lateral 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 circumscribed sphere will be the point of intersection of the perpendiculars that pass through the middle of the edges.
8. A sphere can be inscribed in the 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 vertex 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 a polyhedron lies at the base of the pyramid around which a circle can be described (a necessary and sufficient condition). The center of the sphere will be the point of intersection of the 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 into a pyramid if the bisector planes of the inner dihedral corners of the pyramid intersect at one point (a necessary and sufficient condition). This point will be the center of the sphere.
Connection of a pyramid with a cone
A cone is called inscribed in a pyramid if their tops coincide and the base of the cone is inscribed in the base of the pyramid.
A cone can be inscribed into a pyramid if the apothems of the pyramid are equal to each other.
A cone is called circumscribed around a pyramid if their tops coincide, and the base of the cone is circumscribed around the base of the pyramid.
A cone can be described around the pyramid if all the side edges of the pyramid are equal to each other.
Connection of a pyramid with a cylinder
A pyramid is called 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 described around a pyramid if a circle can be described around the base of the pyramid.
A tetrahedron has four faces and four vertices and six edges, where any two edges do not have common vertices but do not touch.
Each vertex consists of three faces and edges that form triangular corner.
The segment connecting the vertex of the tetrahedron with the center of the opposite face is called median tetrahedron(GM).
Bimedian is the segment connecting the midpoints of opposite edges that are not in contact (KL).
All bimedians and medians of the tetrahedron meet at one point (S). In this case, the bimedians are divided in half, and the medians in the ratio of 3: 1, starting from the top.
Definition. Acute-angled pyramid- this is a pyramid in which the apothem is more than half the length of the side of the base.
Definition. Obtuse pyramid- this is a pyramid in which the apothem is less than half the length of the side of the base.
Definition. Regular tetrahedron- a tetrahedron in which all four faces are equilateral triangles. It is one of the five regular polygons. V regular tetrahedron all dihedral angles (between faces) and trihedral angles (at apex) are equal.
Definition. Rectangular tetrahedron is called a tetrahedron with a right angle between three edges at the vertex (the edges are perpendicular). Three faces form rectangular triangular corner and the faces are right-angled triangles, and the base is an arbitrary triangle. The apothem of any facet is equal to half of the side of the base on which the apothem falls.
Definition. Equhedral tetrahedron called a tetrahedron in which the side faces are equal to each other, and the base is a regular triangle. For such a tetrahedron, the faces are isosceles triangles.
Definition. Orthocentric tetrahedron is called a tetrahedron in which all the heights (perpendiculars) that are lowered from the top to the opposite face intersect at one point.
Definition. Star pyramid is called a polyhedron whose base is a star.
Triangular pyramid is called a polyhedron at the base of which a regular triangle lies.
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 area of the lateral surface 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 lateral surface area triangular pyramid:
where p is the perimeter of the base, in which all sides 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.
Problem: 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 side 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 triangular pyramid area formula to calculate the total 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 the base of a triangular pyramid may be different. Any calculation of parameters for a given figure is allowed, but most often it is not required. Consider an example of calculating the area of the base of a triangular pyramid.
Problem: 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. Let's substitute the data into the formula: 
Quite often it is required 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. 
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.
To begin with, we find the area of the lateral surface using the already known formula. Let's calculate the perimeter:
We 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, one should not forget that a regular triangle lies at the base and many elements of this polyhedron are equal to each other.
The pyramid at the base of which lies regular hexagon, and the sides are formed by regular triangles, called hexagonal.
This polyhedron has many properties:
- All sides and corners of the base are equal to each other;
- All the edges and dihedrals of the coal of the pyramid 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 the correct hexagonal pyramid the standard formula for the lateral surface area of a hexagonal pyramid is applied:
where P is the perimeter of the base, a is the length of the pyramid apothem. In most cases, you can calculate the lateral 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. 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, let's find the perimeter of the base. Since the pyramid is regular, there is a regular hexagon at its base. This means that all its sides are equal, and the perimeter is calculated by the formula:
We substitute the data into 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 base area. 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, based on the conditions from the previous example, from which we know that the side of the base b = 3 cm. 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 sweep: 
Let's consider an example of calculating the area of a hexagonal pyramid.
Let a pyramid be given, at the base of which there is a regular hexagon with side b = 4 cm. The apothem of a given polyhedron is equal to a = 6 cm. Find the total area.
We know that the total area consists of the base and side sweep areas. Therefore, first we will find them. Let's calculate the perimeter:
Now let's find the area of the lateral surface: 
Next, we calculate the area of the base in which the regular hexagon lies: 
Now we can add the resulting results:
When preparing for the exam in mathematics, students have to systematize their knowledge of algebra and geometry. I would like to combine all the 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 with the side faces is clear, 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 shape: from an arbitrary triangle to an n-gon. And this base, in addition to the difference in the number of angles, can be a correct figure or an incorrect one. In the USE tasks of interest to schoolchildren, only tasks with correct figures at the base are encountered. Therefore, we will only talk about them.
Regular triangle
That is, equilateral. The one in which all sides are equal and marked with 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 the side again:
Arbitrary regular n-gon
The side of the polygon has the same symbol. 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 foundation lies correct figure, then 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 the 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.
Square isosceles triangle calculated by a formula in which half of 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 the lateral surface area looks like this:
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 plane angle at its apex (α) are given. Then it is supposed to use the following formula to calculate the lateral area of the pyramid:
S = n / 2 * in 2 sin α .
Problem number 1
Condition. Find the total area of the pyramid if at its base it lies with a side of 4 cm, and the apothem has a value of √3 cm.
Solution. You need to start it by calculating the perimeter of the base. Since this is a regular triangle, 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 a triangle at the base, you get the following area value: (4 2 * √3) / 4 = 4√3 cm 2.
To determine the entire area, you need to add the two resulting values: 6√3 + 4√3 = 10√3 cm 2.
Answer. 10√3 cm 2.
Problem number 2
Condition... There is a regular quadrangular pyramid. The length of the side of the base is 7 mm, the lateral rib is 16 mm. It is necessary to find out its surface area.
Solution. Since the polyhedron is quadrangular and regular, there is a square at its base. Having learned the areas of the base and side faces, it will be possible to calculate the area of the 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 need to calculate the half-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 number 3
Condition... It is necessary to calculate the area of a regular quadrangular pyramid. The side of the square is known in it - 6 cm and the height - 4 cm.
Solution. The easiest way is to use the formula with the product of the perimeter and apothem. The first value is easy to find. The second is a little more complicated.
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 sought apothem (hypotenuse 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 number 4
Condition. The correct side is given. The sides of its base are 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 №2. Only there was given a pyramid with a square at the base, and now it is a hexagon.
The first step is to calculate the area of the base according to 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 the isosceles triangle, which is the side face. (22 + 61 * 2): 2 = 72 cm. It remains to calculate the area of each such triangle using Heron's formula, and then multiply it by six and add it to the one that turned out 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 fold them to find out the entire surface: 5217.47 ~ 5217 cm 2.
Answer. The base is 726√3 cm 2, the lateral surface is 3960 cm 2, the whole area is 5217 cm 2.
