What an apothem for a regular triangular pyramid. What is the apothem for polygon and pyramid? Apothem of a regular quadrangular pyramid. Pyramid Height Base Property

A pyramid is a spatial polyhedron, or a polyhedron, which is found in geometric problems. The main properties of this figure are its volume and surface area, which are calculated from the knowledge of any two of its linear characteristics. One of these characteristics is the apothem of the pyramid. It will be discussed in the article.

figure pyramid

Before giving the definition of the apothem of the pyramid, let's get acquainted with the figure itself. The pyramid is a polyhedron, which is formed by one n-gonal base and n triangles that make up the side surface of the figure.

Every pyramid has a vertex - the junction point of all triangles. The perpendicular drawn from this vertex to the base is called the height. If the height intersects the base in the geometric center, then the figure is called a straight line. A straight pyramid with an equilateral base is called a regular pyramid. The figure shows a pyramid with a hexagonal base, which is viewed from the side of the face and edge.

Apothem of the right pyramid

It is also called apotema. It is understood as a perpendicular drawn from the top of the pyramid to the side of the base of the figure. By definition, this perpendicular corresponds to the height of the triangle that forms the side face of the pyramid.

Since we are considering a regular pyramid with an n-gonal base, then all n apothems for it will be the same, since such are the isosceles triangles of the lateral surface of the figure. Note that identical apothems are a property of a regular pyramid. For a figure of a general type (oblique with an irregular n-gon), all n apothems will be different.

Another property of the apothem of a regular pyramid is that it is simultaneously the height, median and bisector of the corresponding triangle. This means that she divides it into two identical right triangles.

and formulas for determining its apothem

In any regular pyramid, important linear characteristics are the length of the side of its base, the side edge b, the height h and the apothem h b. These quantities are related to each other by the corresponding formulas, which can be obtained by drawing a pyramid and considering the necessary right triangles.

A regular triangular pyramid consists of 4 triangular faces, and one of them (the base) must be equilateral. The rest are isosceles in the general case. The apothem of a triangular pyramid can be determined in terms of other quantities using the following formulas:

h b \u003d √ (b 2 - a 2 / 4);

h b \u003d √ (a 2 / 12 + h 2)

The first of these expressions is valid for a pyramid with any correct base. The second expression is characteristic only for a triangular pyramid. It shows that the apothem is always greater than the height of the figure.

The apothem of a pyramid should not be confused with that of a polyhedron. In the latter case, the apothem is a perpendicular segment drawn to the side of the polyhedron from its center. For example, the apothem of an equilateral triangle is √3/6*a.

Apothem task

Let a regular pyramid with a triangle at the base be given. It is necessary to calculate its apothem if it is known that the area of ​​this triangle is 34 cm 2, and the pyramid itself consists of 4 identical faces.

In accordance with the condition of the problem, we are dealing with a tetrahedron consisting of equilateral triangles. The formula for the area of ​​one face is:

From where we get the length of side a:

To determine the apothem h b, we use the formula containing the side edge b. In the case under consideration, its length is equal to the length of the base, we have:

h b \u003d √ (b 2 - a 2 / 4) \u003d √ 3 / 2 * a

Substituting the value of a through S, we get the final formula:

h b = √3/2*2*√(S/√3) = √(S*√3)

We have obtained a simple formula in which the apothem of a pyramid depends only on the area of ​​its base. If we substitute the value S from the condition of the problem, we get the answer: h b ≈ 7.674 cm.

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 the 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.

Theoretical materials and formulas, see the chapter "Regular Pyramid".

Task

Solution.

Since the pyramid is correct, consider the following:

• The height of the pyramid is projected onto the center of the base
• The center of the base of a regular pyramid according to the condition of the problem is an equilateral triangle
• The center of an equilateral triangle is both the center of the inscribed circle and the circumscribed circle.
• The height of the pyramid forms a right angle with the plane of the base

Since the apothem of a regular pyramid forms a right triangle together with the height of the pyramid, we use the sine theorem to find the height. In addition, let's take into account:

• The first leg of the right triangle under consideration is the height, the second leg is the radius of the inscribed circle (in a regular triangle, the center is both the center of the inscribed and circumscribed circles), the hypotenuse is the apothem of the pyramid
• The third angle of a right-angled triangle is 30 degrees (the sum of the angles of a triangle is 180 degrees, the angle of 60 degrees is given by the condition, the second angle is a right angle according to the properties of the pyramid, the third is 180-90-60 = 30)
• the sine of 30 degrees is 1/2
• the sine of 60 degrees is equal to the square root of three
• the sine of 90 degrees is 1

At the base of the pyramid lies a regular triangle, the area of ​​\u200b\u200bwhich can be found by the formula:
S of an equilateral triangle = 3√3 r 2 .
S = 3√3 2 2 .
S = 12√3.

Now find the volume of the pyramid:
V = 1/3 Sh
V = 1/3 * 12√3 * 2√3
V \u003d 24 cm 3.

Answer: 24 cm3.

Task

Solution .

Since the pyramid is regular, then at its base lies a regular quadrilateral - a square. In addition, the height of the pyramid is projected into the center of the square. Thus, the leg of a right triangle, which is formed by the apothem of the pyramid, the height and the segment connecting them is equal to half the length of the base of a regular quadrangular pyramid.

From where, according to the Pythagorean theorem, the length of the apothem will be found from the equation:

72 + 242 = x2
x2 = 625
x=25

Answer: 25 cm

apothem apothem

(from the Greek apotíthēmi - I postpone), 1) a segment (as well as its length) of a perpendicular A, dropped from the center of a regular polygon to any of its sides. 2) In the correct pyramid, the apothem is the height A side edge.

APOPHEMA (Greek apothema - something postponed),
1) a segment (as well as its length) of the perpendicular a, dropped from the center of a regular polygon to any of its sides.
2) In a regular pyramid, apothem is the height of the side face.

encyclopedic Dictionary. 2009 .

See what "apothem" is in other dictionaries:

See APOTEM. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. APOTHEMA, see APOTHEMA. Dictionary of foreign words included in the Russian language. Pavlenkov F., 1907 ... Dictionary of foreign words of the Russian language

- (from Greek apotithemi I postpone) ..1) a segment (as well as its length) of the perpendicular a, lowered from the center of a regular polygon to any of its sides2)] In a regular pyramid, apothem is the height of the side face ... Big Encyclopedic Dictionary

Exist., number of synonyms: 3 apotema (2) length (10) perpendicular (4) Dictionary ... Synonym dictionary

APOTHEM- (1) the length of the perpendicular dropped from the center of a circle circumscribed around a regular polygon to any of its sides; (2) the height of the side face of a regular pyramid; (3) the height of the trapezoid, which is the side face of a regular truncated ... ... Great Polytechnic Encyclopedia

- (from the Greek apotithçmi I put aside) 1) the length of the perpendicular dropped from the center of a regular polygon to any of its sides (Fig. 1); 2) in a regular pyramid A. the height a of its lateral face (Fig. 2). Rice. 1 to… … Great Soviet Encyclopedia

- (from the Greek apotfthemi I postpone) 1) a segment (as well as its length) of the perpendicular a, lowered from the center of a regular polygon to any of its sides. 2) In a regular pyramid A., the height a of the side face (see figure). To Art. Apothem... Big encyclopedic polytechnic dictionary

The length of a perpendicular dropped from the center of a regular polygon to one of its sides; the apothem is equal to the radius of the circle inscribed in the given polygon. A. was also called the inclined side of the cone ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron

- (from the Greek apotithemi I postpone), 1) a segment (as well as its length) of the perpendicular a, lowered from the center of a regular polygon to any of its sides. 2) In a regular pyramid A. the height a of the side face ... Natural science. encyclopedic Dictionary

Apothem, apothem, apothem, apothem, apothem, apothem, apothem, apothem, apothem, apothem, apothem, apothem, apothem (

• apothem- the height of the side face of a regular pyramid, which is drawn from its top (in addition, the apothem is the length of the perpendicular, which is lowered from the middle of a regular polygon to 1 of its sides);
• side faces (ASB, BSC, CSD, DSA) - triangles that converge at the top;
• side ribs ( AS , BS , CS , D.S. ) - common sides of the side faces;
• top of the pyramid (v. S) - a point that connects the side edges and which does not lie in the plane of the base;
• height ( SO ) - a segment of the perpendicular, which is drawn through the top of the pyramid to the plane of its base (the ends of such a segment will be the top of the pyramid and the base of the perpendicular);
• diagonal section of a pyramid- section of the pyramid, which passes through the top and the diagonal of the base;
• base (ABCD) is a polygon to which the top of the pyramid does not belong.

pyramid properties.

1. When all side edges are the same size, then:

• near the base of the pyramid it is easy to describe a circle, while the top of the pyramid will be projected into the center of this circle;
• side ribs form equal angles with the base plane;
• in addition, the converse is also true, i.e. when the side edges form equal angles with the base plane, or when a circle can be described near the base of the pyramid and the top of the pyramid will be projected into the center of this circle, then all the side edges of the pyramid have the same size.

2. When the side faces have an angle of inclination to the plane of the base of the same value, then:

• near the base of the pyramid, it is easy to describe a circle, while the top of the pyramid will be projected into the center of this circle;
• the heights of the side faces are of equal length;
• the area of ​​the side surface is ½ the product of the perimeter of the base and the height of the side face.

3. A sphere can be described near the pyramid if the base of the pyramid is a polygon 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 that pass through the midpoints of the edges of the pyramid perpendicular to them. From this theorem we conclude that a sphere can be described both around any triangular and around any regular pyramid.

4. A sphere can be inscribed in a pyramid if the bisector planes of the internal dihedral angles of the pyramid intersect at the 1st point (a necessary and sufficient condition). This point will become the center of the sphere.

The simplest pyramid.

According to the number of corners of the base of the pyramid, they are divided into triangular, quadrangular, and so on.

The pyramid will triangular, quadrangular, and so on, when the base of the pyramid is a triangle, a quadrilateral, and so on. A triangular pyramid is a tetrahedron - a tetrahedron. Quadrangular - pentahedron and so on.