Formula for calculating the lateral surface of a pyramid. Pyramid

The area of ​​the lateral surface of an arbitrary pyramid is equal to the sum of the areas of its lateral faces. It makes sense to give a special formula for expressing this area in the case of a regular pyramid. So, let a regular pyramid be given, at the base of which lies a regular n-gon with a side equal to a. Let h be the height of the side face, also called apothema pyramids. The area of ​​one side face is 1/2ah, and the entire side surface of the pyramid has an area equal to n/2ha. Since na is the perimeter of the base of the pyramid, we can write the found formula as follows:

Lateral surface area of a regular pyramid is equal to the product of its apothem by half the perimeter of the base.

Concerning total surface area, then simply add the area of ​​\u200b\u200bthe base to the side.

Inscribed and circumscribed sphere and ball. It should be noted that the center of the sphere inscribed in the pyramid lies at the intersection of the bisector planes of the internal dihedral angles of the pyramid. The center of the sphere described near the pyramid lies at the intersection of planes passing through the midpoints of the edges of the pyramid and perpendicular to them.

Truncated pyramid. If the pyramid is cut by a plane parallel to its base, then the part enclosed between the cutting plane and the base is called truncated pyramid. The figure shows a pyramid, discarding its part lying above the cutting plane, we get a truncated pyramid. It is clear that the small pyramid to be discarded is homothetic to the large pyramid with the center of the homothety at the apex. The similarity coefficient is equal to the ratio of heights: k=h 2 /h 1 , or side ribs, or other corresponding linear dimensions of both pyramids. We know that the areas of similar figures are related as squares of linear dimensions; so the areas of the bases of both pyramids (i.e. spare the bases of the truncated pyramid) are related as

Here S 1 is the area of ​​the lower base, and S 2 is the area of ​​the upper base of the truncated pyramid. The side surfaces of the pyramids are in the same ratio. There is a similar rule for volumes.

Volumes of similar bodies are related as cubes of their linear dimensions; for example, the volumes of the pyramids are related as the products of their heights by the area of ​​​​the bases, from which our rule immediately follows. It has absolutely general character and follows directly from the fact that the volume always has the dimension of the third power of length. Using this rule, we derive a formula expressing the volume of a truncated pyramid in terms of the height and areas of the bases.

Let a truncated pyramid with height h and base areas S 1 and S 2 be given. If we imagine that it is extended to the full pyramid, then the similarity coefficient of the full pyramid and the small pyramid can be easily found as the root of the S 2 /S 1 ratio. The height of the truncated pyramid is expressed as h = h 1 - h 2 = h 1 (1 - k). Now we have for the volume of the truncated pyramid (V 1 and V 2 denote the volumes of the full and small pyramids)

truncated pyramid volume formula

We derive the formula for the area S of the lateral surface of a regular truncated pyramid through the perimeters P 1 and P 2 of the bases and the length of the apothem a. We argue in exactly the same way as when deriving the formula for volume. We supplement the pyramid with the upper part, we have P 2 \u003d kP 1, S 2 \u003d k 2 S 1, where k is the similarity coefficient, P 1 and P 2 are the perimeters of the bases, and S 1 and S 2 are the horses of the side surfaces of the entire resulting pyramid and its top, respectively. For the lateral surface we find (a 1 and a 2 - apothems of the pyramids, a \u003d a 1 - a 2 \u003d a 1 (1-k))

formula for the lateral surface area of ​​a regular truncated pyramid

• Task 1. Find the total surface area of ​​the pyramid
• Task 2. Find the area of ​​the lateral surface of a regular triangular pyramid

Task 1. Find the total surface area of ​​a regular pyramid

Solution.

At the base of a regular triangular pyramid lies an equilateral triangle.
Therefore, to solve the problem, we use the properties of a regular triangle:

We know the height of the triangle, from where we can find its area.
h = √3/2a
a = h / (√3/2)
a = 3 / (√3/2)
a = 6 / √3

From where the area of ​​​​the base will be equal to:
S = √3/4 a 2
S = √3/4 (6 / √3) 2
S = 3√3

In order to find the area of ​​the side face, we calculate the height KM. The OKM angle, according to the problem statement, is 45 degrees.
In this way:
OK / MK = cos 45
We use the table of values ​​of trigonometric functions and substitute known values.

OK / MK = √2/2

We take into account that OK is equal to the radius of the inscribed circle. Then
OK = √3/6 a
OK = √3/6 * 6/√3 = 1

Then
OK / MK = √2/2
1 / MK = √2/2
MK = 2/√2

The area of ​​the side face is then equal to half the product of the height and the base of the triangle.
Sside = 1/2 (6 / √3) (2/√2) = 6/√6

Thus, the total surface area of ​​the pyramid will be equal to
S = 3√3 + 3 * 6/√6
S = 3√3 + 18/√6

Answer: 3√3 + 18/√6

Task 2. Find the lateral surface area of ​​a regular pyramid

Solution.

Since the base of a regular triangular pyramid is an equilateral triangle, then AO is the radius of the circumscribed circle around the base.
(It follows from)

The radius of a circle circumscribed around an equilateral triangle is found from its properties

Whence the length of the edges of a regular triangular pyramid will be equal to:
AM 2 = MO 2 + AO 2
the height of the pyramid is known by the condition (10 cm), AO = 16√3/3
AM 2 = 100 + 256/3
AM = √(556/3)

Each side of the pyramid is an isosceles triangle. Square isosceles triangle find from the first formula below

S = 1/2 * 16 sqrt((√(556/3) + 8) (√(556/3) - 8))
S = 8 sqrt((556/3) - 64)
S = 8 sqrt(364/3)
S = 16 sqrt(91/3)

Since all three faces of a regular pyramid are equal, the lateral surface area will be equal to
3S = 48√(91/3)

Answer: 48 √(91/3)

Task 3. Find the total surface area of ​​a regular pyramid

The side of a regular triangular pyramid is 3 cm and the angle between the side face and the base of the pyramid is 45 degrees. Find the total surface area of ​​the pyramid.

Solution.
Since the pyramid is regular, it has an equilateral triangle at its base. So the area of ​​the base is


So = 9 * √3/4

In order to find the area of ​​the side face, we calculate the height KM. The OKM angle, according to the problem statement, is 45 degrees.
In this way:
OK / MK = cos 45
Let's use

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 is a pyramid in which the base is 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 everyone side ribs 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 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 a side faces 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 edges 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.

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What shape do we call a pyramid? First, 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 form of triangles converging at one common vertex. Now, having dealt with the term, let's find out how to find the surface area of ​​the pyramid.

It is clear that the surface area of ​​such 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 the pyramid

The choice of the calculation formula depends on the shape of the polygon lying at the base of our pyramid. It can be correct, that is, with sides of the same length, or incorrect. Let's consider both options.

At the base is a regular polygon

From school course known:

• 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 times the square root of three.

But there is also a general formula for calculating the area of ​​any regular polygon (Sn): you need to multiply the value of 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 .

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.

Side 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 sides. There are also 2 options here.

1. Let us have an arbitrary pyramid, i.e. one whose base is an irregular polygon. Then you should calculate separately the area of ​​each face and add the results. Since the sides of the pyramid, by definition, can only be triangles, the calculation is based on the formula mentioned above: 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 in its center. Then, to calculate the area of ​​the side surface (Sb), it is enough to find half the product of the perimeter of the base polygon (P) and the height (h) of the side (the same for all faces): Sb \u003d 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 calculate algebraically the surface areas of several pyramids.

Surface area of ​​a triangular pyramid

At the base of such a pyramid is a triangle. According to the formula So \u003d 1 / 2a * h, we find the area of ​​\u200b\u200bthe base. We apply the same formula to find the area of ​​each face of the pyramid, also having 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 \u003d S1 + S2 + S3. Adding the areas of the sides and base, we get the total surface area of ​​the desired pyramid: Sp \u003d So + Sb.

Surface area of ​​a quadrangular pyramid

The lateral surface area is the sum of 4 terms: Sb \u003d S1 + S2 + S3 + S4, each of which is calculated using the triangle area formula. And the area of ​​\u200b\u200bthe base will have to be sought, depending on the shape of the quadrangle - correct 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.