The lateral surface area of ​​a straight pyramid is equal. Lateral surface area of ​​different pyramids

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 general formula, to calculate 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.

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:

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 happen 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 – 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.

In a regular triangular pyramid SABC R- middle of the rib AB, S- top.
It is known that SR = 6, and the lateral surface area is equal to 36 .
Find the length of the segment B.C..

Let's make a drawing. In a regular pyramid, the side faces are isosceles triangles.

Line segment S.R.- the median lowered to the base, and therefore the height of the side face.

The lateral surface area of ​​a regular triangular pyramid is equal to the sum of the areas
three equal side faces S side = 3 S ABS. From here S ABS = 36: 3 = 12- area of ​​the face.

The area of ​​a triangle is equal to half the product of its base and height
S ABS = 0.5 AB SR. Knowing the area and height, we find the side of the base AB = BC.
12 = 0.5 AB 6
12 = 3 AB
AB = 4

Answer: 4

You can approach the problem from the other end. Let the base side AB = BC = a.
Then the area of ​​the face S ABS = 0.5 AB SR = 0.5 a 6 = 3a.

The area of ​​each of the three faces is equal to 3a, the area of ​​the three faces is equal 9a.
According to the conditions of the problem, the area of ​​the lateral surface of the pyramid is 36.
S side = 9a = 36.
From here a = 4.

Definition. Side edge- this is a triangle in which one angle lies at the top of the pyramid, and the opposite side coincides with the side of the base (polygon).

Definition. Side ribs- these are the common sides of the side faces. A pyramid has as many edges as the angles of a polygon.

Definition. Pyramid height- this is a perpendicular lowered from the top to the base of the pyramid.

Definition. Apothem- this is a perpendicular to 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 a 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 descends to the center of the base.

Volume and surface area of ​​the pyramid

Formula. Volume of the pyramid through base area and height:

Properties of the pyramid

If all the side edges are equal, then a circle can be drawn around the base of the pyramid, and the center of the base coincides with the center of the circle. Also, a perpendicular dropped from the top passes through the center of the base (circle).

If all the side edges are equal, then they are inclined to the plane of the base at the same angles.

The lateral 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 plane of the base at the same 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 plane of the base 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 are inclined at equal angles 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 intersection point of the perpendiculars that pass through the middle of the edges.

8. You can fit a sphere into a pyramid. The center of the inscribed sphere will be the point of intersection 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 plane 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 between the pyramid and the sphere

A sphere can be described around a pyramid when at the base of the pyramid there is a polyhedron around which a circle can be described (a necessary and sufficient condition). The center of the sphere will be the intersection point of planes passing perpendicularly through the midpoints of the side edges of the pyramid.

It is always possible to describe a sphere 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.

Connection of a pyramid with a cone

A cone is said to be 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 to each other.

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 the lateral edges of the pyramid are equal to each other.

Relationship between a pyramid and 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.

Definition. Truncated pyramid (pyramidal prism) is a polyhedron that is located between the base of the pyramid and the section plane parallel to the base. Thus a pyramid has a larger base and a smaller base that is similar to the larger one. The side faces are trapezoidal.

Definition. Triangular pyramid (tetrahedron) is a pyramid in which three faces and the base are arbitrary triangles.

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 angle.

The segment connecting the vertex of a tetrahedron with the center of the opposite face is called median of the tetrahedron(GM).

Bimedian 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 are divided in a ratio of 3:1 starting from the top.

Definition. Slanted pyramid is a pyramid in which one of the edges forms an obtuse angle (β) with the base.

Definition. Rectangular pyramid is a pyramid in which one of the side faces is perpendicular to the base.

Definition. Acute angled pyramid- a pyramid in which the apothem is more than half the length of the side of the base.

Definition. Obtuse pyramid- 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. In a regular tetrahedron, all dihedral angles (between faces) and trihedral angles (at the vertex) are equal.

Definition. Rectangular tetrahedron is called a tetrahedron in which there is a right angle between three edges at the apex (the edges are perpendicular). Three faces form rectangular triangular 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 is called a tetrahedron whose side faces are equal to each other, and the base is a regular triangle. Such a tetrahedron has faces that 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 called a polyhedron whose base is a star.

Definition. Bipyramid- a polyhedron consisting of two different pyramids (pyramids can also be cut off) having common ground, and the vertices lie on opposite sides of the base plane.