The area of ​​the lateral surface of the pyramid. How to calculate the area of ​​\u200b\u200bthe pyramid: base, lateral and full

- This is a polyhedral figure, at the base of which lies a polygon, and the remaining faces are represented by triangles with a common vertex.

If the base is a square, then a pyramid is called quadrangular, if the triangle is triangular. The height of the pyramid is drawn from its top perpendicular to the base. Also used to calculate the area apothem is the height of the side face lowered from its vertex.
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 ​​\u200b\u200bthe pyramid is calculated through the perimeter of the base and the apothem:

Consider an example of calculating the area of ​​the lateral surface of a pyramid.

Let a pyramid with base ABCDE and apex F be given. 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 faces of the base are equal, then the perimeter of the pentagon will be equal to:
Now you can find the side area of ​​the pyramid:

Area of ​​a regular triangular pyramid

A regular triangular pyramid consists of a base in which a regular triangle lies 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 many ways. You can apply the usual formula for calculating through the perimeter and apothem, or you can find the area of ​​\u200b\u200bone face and multiply it by three. Since the face of the 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. Consider an example of calculating the lateral surface area of ​​a regular triangular pyramid.

Given a pyramid with an apothem a = 4 cm and a 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 ​​in the formula:
Since in a regular pyramid all 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:

The area of ​​the truncated pyramid

Truncated A pyramid is a polyhedron formed by a pyramid and its 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:

A parallelepiped is a quadrangular prism with a parallelogram at its base. There are ready-made formulas for calculating the lateral and total surface area of ​​\u200b\u200bthe figure, for which only the lengths of three dimensions of the parallelepiped are needed.

How to find the lateral surface area of ​​a cuboid

It is necessary to distinguish between a rectangular and a right parallelepiped. The base of a straight figure can be any parallelogram. The area of ​​such a figure must be calculated using other formulas.

The sum S of the side faces of a cuboid is calculated using the simple formula P*h, where P is the perimeter and h is the height. The figure shows that the opposite faces of a rectangular parallelepiped are equal, and the height h coincides with the length of the edges perpendicular to the base.

Surface area of ​​a cuboid

The total area of ​​the figure consists of the side and the area of ​​2 bases. How to find the area of ​​a rectangular parallelepiped:

Where a, b and c are the dimensions of the geometric body.
The formulas described are easy to understand and useful in solving many geometry problems. An example of a typical task is shown in the following image.

When solving problems of this kind, it should be remembered that the base of a quadrangular prism is chosen arbitrarily. If we take a face with dimensions x and 3 as the base, then the values ​​of Sside will be different, and Stot will remain 94 cm2.

Cube surface area

A cube is a rectangular parallelepiped with all 3 dimensions equal. In this regard, the formulas for the total and lateral area of ​​a cube differ from the standard ones.

The perimeter of the cube is 4a, therefore, Sside = 4*a*a = 4*a2. These expressions are not required for memorization, but significantly speed up the solution of tasks.

Pyramid- one of the varieties of a polyhedron formed from polygons and triangles that lie at the base and are its faces.

Moreover, at the top of the pyramid (i.e. at one point), all faces are combined.

In order to calculate the area of ​​the pyramid, it is worth determining that its lateral surface consists of several triangles. And we can easily find their areas using

various formulas. Depending on what data of triangles we know, we are looking for their area.

We list some formulas with which you can find the area of ​​triangles:

1. S = (a*h)/2 . In this case, we know the height of the triangle h , which is lowered to the side a .
2. S = a*b*sinβ . Here the sides of the triangle a , b , and the angle between them is β .
3. S = (r*(a + b + c))/2 . Here the sides of the triangle a, b, c . The radius of a circle that is inscribed in a triangle is r .
4. S = (a*b*c)/4*R . The radius of the circumscribed circle around the triangle is R .
5. S = (a*b)/2 = r² + 2*r*R . This formula should only be applied if the triangle is a right triangle.
6. S = (a²*√3)/4 . We apply this formula to an equilateral triangle.

Only after we calculate the areas of all the triangles that are the faces of our pyramid, can we calculate the area of ​​\u200b\u200bits lateral surface. To do this, we will use the above formulas.

In order to calculate the area of ​​​​the lateral surface of the pyramid, no difficulties arise: you need to find out the sum of the areas of all triangles. Let's express this with the formula:

Sp = ΣSi

Here Si is the area of ​​the first triangle, and S P is the area of ​​the lateral surface of the pyramid.

Let's look at an example. Given a regular pyramid, its lateral faces are formed by several equilateral triangles,

« Geometry is the most powerful tool for the refinement of our mental faculties.».

Galileo Galilei.

and the square is the base of the pyramid. Moreover, the edge of the pyramid has a length of 17 cm. Let's find the area of ​​the lateral surface of this pyramid.

We reason like this: we know that the faces of the pyramid are triangles, they are equilateral. We also know what is the length of the edge of this pyramid. It follows that all triangles have equal sides, their length is 17 cm.

To calculate the area of ​​each of these triangles, you can use the following formula:

S = (17²*√3)/4 = (289*1.732)/4 = 125.137 cm²

Since we know that the square lies at the base of the pyramid, it turns out that we have four equilateral triangles. This means that the area of ​​the lateral surface of the pyramid can be easily calculated using the following formula: 125.137 cm² * 4 = 500.548 cm²

Our answer is the following: 500.548 cm² - this is the area of ​​the lateral surface of this pyramid.

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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.

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

Definition. Triangular pyramid (tetrahedron)- this 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 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. inclined 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 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.

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