A pyramid is a spatial polyhedron, or polyhedron, that is found in geometric problems. The main properties of this figure are its volume and surface area, which are calculated from 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.
Pyramid figure
Before giving the definition of the apothem of a pyramid, let's get acquainted with the figure itself. A pyramid is a polyhedron, which is formed by one n-gonal base and n triangles that make up the lateral surface of the figure.
Every pyramid has a vertex - the point of connection of all triangles. The perpendicular drawn from this vertex to the base is called the height. If the height intersects the base at the geometric center, then the figure is called a straight line. A straight pyramid with an equilateral base is called regular. The figure shows a pyramid with a hexagonal base, viewed from the sides and edges.
Apothem of a regular pyramid
It is also called apothem. It is understood as a perpendicular drawn from the top of the pyramid to the side of the base of the figure. By its 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 these 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 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 it divides it into two identical right triangles.
and formulas for determining its apothem
In any regular pyramid, the important linear characteristics are the length of the side of its base, the lateral 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 = √(b 2 - a 2 /4);
h b = √(a 2 /12 + h 2)
The first of these expressions is true for a pyramid with any regular base. The second expression is typical exclusively 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, an 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 calculation problem
Let us be given a regular pyramid with a triangle at the base. 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 conditions of the problem, we are dealing with a tetrahedron consisting of equilateral triangles. The formula for the area of one face is:
Where do we get the length of side a:
To determine the apothem h b, we use a formula containing the lateral edge b. In the case under consideration, its length is equal to the length of the base, we have:
h b = √(b 2 - a 2 /4) = √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 of S from the problem conditions, we get the answer: h b ≈ 7.674 cm.
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 edges are equal when they form equal angles with the plane of the base 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 into 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.
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. 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 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 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.
To successfully solve problems in geometry, you must clearly understand the terms that this science uses. For example, these are “straight”, “plane”, “polyhedron”, “pyramid” and many others. In this article we will answer the question of what an apothem is.
Double use of the term "apothem"
In geometry, the meaning of the word "apothema" or "apothema", as it is also called, depends on the object to which it is applied. There are two fundamentally different classes of figures in which it is one of their characteristics.
First of all, these are flat polygons. What is an apothem for a polygon? This is the height drawn from the geometric center of the figure to any of its sides.
To make it clearer what we mean we're talking about, let's look at a specific example. Let's assume that we have a regular hexagon as shown in the figure below.
The symbol l denotes the length of its side, and the letter a denotes the apothem. For a marked triangle, it is not only the height, but also the bisector and the median. It is easy to show that through the side l it can be calculated as follows:
The apothem is defined similarly for any n-gon.
Secondly, these are pyramids. What is an apothem for such a figure? This issue requires more detailed consideration.
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Pyramids and their apothems
First, let's define a pyramid from a geometric point of view. This figure is a three-dimensional body formed by one n-gon (base) and n triangles (sides). The latter are connected at one point, which is called the vertex. The distance from it to the base is the height of the figure. If it falls on the geometric center of the n-gon, then the pyramid is called a straight line. If, in addition, the n-gon has equal angles and sides, then the figure is called regular. Below is an example of a pyramid.
What is an apothem for such a figure? This is the perpendicular that connects the sides of the n-gon to the vertex of the figure. Obviously, it represents the height of the triangle, which is the side of the pyramid.
Apothem is convenient to use when solving geometric problems with regular pyramids. The fact is that for them all the side faces are isosceles triangles equal to each other. The last fact means that all n apothems are equal, so for a regular pyramid we can talk about one and only such straight line.
Apothem of a regular quadrangular pyramid
Perhaps the most obvious example of this figure will be the famous first wonder of the world - the Pyramid of Cheops. She is located in Egypt.
For any such figure with a regular n-gonal base, we can give formulas that allow us to determine its apothem through the length a of the side of the polygon, through the lateral edge b and the height h. Here we write down the corresponding formulas for a straight pyramid with a square base. The apothem h b for it will be equal to:
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h b = √(b 2 - a 2 /4);
h b = √(h 2 + a 2 /4)
The first of these expressions is valid for any regular pyramid, the second - only for a quadrangular one.
Let's show how these formulas can be used to solve the problem.
Geometric problem
Let a straight pyramid with a square base be given. It is necessary to calculate its base area. The apothem of the pyramid is 16 cm, and its height is 2 times the side of the base.
Every schoolchild knows: to find the area of the square, which is the base of the pyramid in question, you need to know its side a. To find it, we use the following formula for apothem:
h b = √(h 2 + a 2 /4)
The meaning of the apothem is known from the conditions of the problem. Since the height h is twice the length of the side a, this expression can be transformed as follows:
h b = √((2*a) 2 + a 2 /4) = a/2*√17 =>
a = 2*h b /√17
The area of a square is equal to the product of its sides. Substituting the resulting expression for a, we have:
S = a 2 = 4/17*h b 2
It remains to substitute the apothem value from the problem conditions into the formula and write down the answer: S ≈ 60.2 cm 2.
Read also:
- apothem- the height of the side face of a regular pyramid, which is drawn from its vertex (in addition, the apothem is the length of the perpendicular, which is lowered from the middle of the regular polygon to one of its sides);
- side faces (ASB, BSC, CSD, DSA) - triangles that meet at the vertex;
- lateral ribs ( AS , B.S. , C.S. , D.S. ) — common sides of the side faces;
- top of the pyramid (t. S) - a point that connects the side ribs and which does not lie in the plane of the base;
- height ( SO ) - a perpendicular segment 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 the pyramid- a section of the pyramid that passes through the top and the diagonal of the base;
- base (ABCD) - a polygon that does not belong to the vertex of the pyramid.
Properties of the pyramid.
1. When all side edges are of the same size, then:
- it is easy to describe a circle near the base of the pyramid, and the top of the pyramid will be projected into the center of this circle;
- the side ribs form equal angles with the plane of the base;
- Moreover, the opposite is also true, i.e. when the side ribs form equal angles with the plane of the base, or when a circle can be described around the base of the pyramid and the top of the pyramid will be projected into the center of this circle, it means that all the side edges of the pyramid are the same size.
2. When the side faces have an angle of inclination to the plane of the base of the same value, then:
- it is easy to describe a circle near the base of the pyramid, and 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 equal to ½ the product of the perimeter of the base and the height of the side face.
3. A sphere can be described around a pyramid if at the base of the pyramid there 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 middles 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 into 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.
Based on the number of angles, the base of the pyramid is divided into triangular, quadrangular, and so on.
There will be a pyramid triangular, quadrangular, and so on, when the base of the pyramid is a triangle, a quadrangle, and so on. A triangular pyramid is a tetrahedron - a tetrahedron. Quadrangular - pentagonal and so on.