Before studying questions about this geometric figure and its properties, it is necessary to understand some terms. When a person hears about the pyramid, he imagines huge buildings in Egypt. This is what the simplest ones look like. But they come in different types and shapes, which means that the calculation formula for geometric shapes will be different.
Pyramid - geometric figure, denoting and representing multiple faces. In fact, 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 is of two main types:
- correct;
- truncated.
In the first case, the base is a regular polygon. Here all side 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 section formed parallel to the base.
Terms and notation
Basic terms:
- Regular (equilateral) triangle A figure with three identical angles and equal sides. In this case, all angles are 60 degrees. The figure is the simplest of the regular polyhedra. If this figure lies at the base, then such a polyhedron will be called a regular triangular one. 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 top is formed by a straight line emanating from the top to the base of the pyramid.
- edge is 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.
- cross section- a flat figure formed as a result of dissection. Not to be confused with a section, as 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. This definition is valid only in relation to a regular polyhedron. For example - if it is not a truncated pyramid, then the face will be a triangle. In this case, the height of this triangle will become an apothem.
Area formulas
Find the area of the lateral surface 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 \u200b\u200beach face and add them together.
Depending on what parameters are known, formulas for calculating a square, a trapezoid, an arbitrary quadrangle, etc. may be required. The formulas themselves in different cases will also be different.
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 precisely for such figures. Therefore, the corresponding formulas will be given below. Otherwise, you would have to paint everything on several pages, which will only confuse and confuse.
Basic formula for calculation the lateral surface area of a regular pyramid will look like this:
S \u003d ½ Pa (P is the perimeter of the base, and is the apothem)
Let's consider one of the examples. The polyhedron has a base with segments A1, A2, A3, A4, A5, and they are all 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, it can be found as follows: P \u003d 5 * 10 \u003d 50 cm. Next, we apply the basic formula: S \u003d ½ * 50 * 5 \u003d 125 cm squared.
Lateral surface area of a regular triangular pyramid the easiest to calculate. The formula looks like this:
S =½* ab *3, where a is the apothem, b is the facet 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. Consider an example. Given a figure with an apothem of 5 cm and a base face 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 \u003d 1/2 * (p _01 + p _02) * a, where p_01 and p_02 are the perimeters of the bases, and is the apothem. Consider an example. Suppose, for a quadrangular figure, the dimensions of the sides of the bases are 3 and 6 cm, the apothem is 4 cm.
Here, for starters, you should find the perimeters of the bases: p_01 \u003d 3 * 4 \u003d 12 cm; p_02=6*4=24 cm. It remains to substitute the values into the main formula and get: S =1/2*(12+24)*4=0.5*36*4=72 cm squared.
Thus, it is possible to find the lateral surface area of a regular pyramid of any complexity. Be careful not to confuse these calculations with the total area of the entire polyhedron. And if you still need to do this, it’s enough to calculate the area of \u200b\u200bthe largest base of the polyhedron and add it to the area of \u200b\u200bthe lateral surface of the polyhedron.
Video
To consolidate information on how to find the lateral surface area of different pyramids, this video will help you.
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Students come across the concept of a pyramid long before studying geometry. Blame the famous great Egyptian wonders of the world. Therefore, starting the study of this wonderful polyhedron, most students already clearly imagine it. All of the above sights are in the correct shape. What right pyramid, and what properties it has and will be discussed further.
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Definition
There are many definitions of a pyramid. Since ancient times, it has been very popular.
For example, Euclid defined it as a solid figure, consisting of planes, which, starting from one, converge at a certain point.
Heron provided a more precise formulation. He insisted that it was a figure that has a base and planes in the form of triangles, converging at one point.
Based on the modern interpretation, the pyramid is presented as a spatial polyhedron, consisting of a certain k-gon and k flat triangular figures that have one common point.
Let's take a closer look, What elements does it consist of?
- k-gon is considered the basis of the figure;
- 3-angled figures protrude as the sides of the side part;
- the upper part, from which the side elements originate, is called the top;
- all segments connecting the vertex are called edges;
- if a straight line is lowered from the top to the plane of the figure at an angle of 90 degrees, then its part enclosed in the inner space is the height of the pyramid;
- in any side element to the side of our polyhedron, you can draw a perpendicular, called apothem.
The number of edges is calculated using the formula 2*k, where k is the number of sides of the k-gon. How many faces a polyhedron like a pyramid has can be determined by the expression k + 1.
Important! A regular-shaped pyramid is a stereometric figure whose base plane is a k-gon with equal sides.
Basic properties
Correct pyramid has many properties that are unique to her. Let's list them:
- The base is a figure of the correct form.
- The edges of the pyramid, limiting the side elements, have equal numerical values.
- The side elements are isosceles triangles.
- The base of the height of the figure falls into the center of the polygon, while it is simultaneously the central point of the inscribed and described.
- All side ribs are inclined to the base plane at the same angle.
- All side surfaces have the same angle of inclination with respect to the base.
Thanks to all the listed properties, the performance of element calculations is greatly simplified. Based on the above properties, we pay attention to two signs:
- In the case when the polygon fits into a circle, the side faces will have equal angles with the base.
- When describing a circle around a polygon, all the edges of the pyramid emanating from the vertex will have the same length and equal angles with the base.
The square is based
Regular quadrangular pyramid - a polyhedron based on a square.
It has four side faces, which are isosceles in appearance.
On a plane, a square is depicted, but they are based on all the properties of a regular quadrilateral.
For example, if it is necessary to connect the side of a square with its diagonal, then the following formula is used: the diagonal is equal to the product of the side of the square and the square root of two.
Based on a regular triangle
A regular triangular pyramid is a polyhedron whose base is a regular 3-gon.
If the base is a regular triangle, and the side edges are equal to the edges of the base, then such a figure called a tetrahedron.
All faces of a tetrahedron are equilateral 3-gons. In this case, you need to know some points and not waste time on them when calculating:
- the angle of inclination of the ribs to any base is 60 degrees;
- the value of all internal faces is also 60 degrees;
- any face can act as a base;
- drawn inside the figure are equal elements.
Sections of a polyhedron
In any polyhedron there are several types of sections plane. Often in a school geometry course they work with two:
- axial;
- parallel basis.
An axial section is obtained by intersecting a polyhedron with a plane that passes through the vertex, side edges and axis. In this case, the axis is the height drawn from the vertex. The cutting plane is limited by the lines of intersection with all faces, resulting in a triangle.
Attention! In a regular pyramid, the axial section is an isosceles triangle.
If the cutting plane runs parallel to the base, then the result is the second option. In this case, we have in the context of a figure similar to the base.
For example, if the base is a square, then the section parallel to the base will also be a square, only of a smaller size.
When solving problems under this condition, signs and properties of similarity of figures are used, based on the Thales theorem. First of all, it is necessary to determine the coefficient of similarity.
If the plane is drawn parallel to the base, and it cuts off the upper part of the polyhedron, then a regular truncated pyramid is obtained in the lower part. Then the bases of the truncated polyhedron are said to be similar polygons. In this case, the side faces are isosceles trapezoids. The axial section is also isosceles.
In order to determine the height of a truncated polyhedron, it is necessary to draw the height in an axial section, that is, in a trapezoid.
Surface areas
The main geometric problems that have to be solved in the school geometry course are finding the surface area and volume of a pyramid.
There are two types of surface area:
- area of side elements;
- the entire surface area.
From the title itself it is clear what it is about. The side surface includes only the side elements. From this it follows that to find it, you simply need to add up the areas of the lateral planes, that is, the areas of isosceles 3-gons. Let's try to derive the formula for the area of the side elements:
- The area of an isosceles 3-gon is Str=1/2(aL), where a is the side of the base, L is the apothem.
- The number of side planes depends on the type of the k-gon at the base. For example, a regular quadrangular pyramid has four lateral planes. Therefore, it is necessary to add up the areas of four figures Sside=1/2(aL)+1/2(aL)+1/2(aL)+1/2(aL)=1/2*4a*L. The expression is simplified in this way because the value 4a=POS, where POS is the perimeter of the base. And the expression 1/2 * Rosn is its semi-perimeter.
- So, we conclude that the area of the side elements of a regular pyramid is equal to the product of the semi-perimeter of the base and the apothem: Sside \u003d Rosn * L.
The area of the full surface of the pyramid consists of the sum of the areas of the lateral planes and the base: Sp.p. = Sside + Sbase.
As for the area of \u200b\u200bthe base, here the formula is used according to the type of polygon.
Volume of a regular pyramid is equal to the product of the base plane area and the height divided by three: V=1/3*Sbase*H, where H is the height of the polyhedron.
What is a regular pyramid in geometry
Properties of a regular quadrangular pyramid
Typical geometric problems in the plane and in three-dimensional space are the problems of determining the surface areas of different figures. In this article, we present the formula for the area of the lateral surface of a regular quadrangular pyramid.
What is a pyramid?
Let us give a strict geometric definition of a pyramid. Suppose there is some polygon with n sides and n corners. We choose an arbitrary point in space that will not be in the plane of the specified n-gon, and connect it to each vertex of the polygon. We will get a figure that has some volume, which is called an n-gonal pyramid. For example, let's show in the figure below what a pentagonal pyramid looks like.
Two important elements of any pyramid are its base (n-gon) and top. These elements are connected to each other by n triangles, which in general are not equal to each other. The perpendicular dropped from the top to the base is called the height of the figure. If it intersects the base in the geometric center (coincides with the center of mass of the polygon), then such a pyramid is called a straight line. If, in addition to this condition, the base is a regular polygon, then the entire pyramid is called regular. The figure below shows what regular pyramids look like with triangular, quadrangular, pentagonal, and hexagonal bases.
The surface of the pyramid
Before turning to the question of the area of the lateral surface of a regular quadrangular pyramid, one should dwell in more detail on the concept of the surface itself.
As mentioned above and shown in the figures, any pyramid is formed by a set of faces or sides. One side is the base and n sides are triangles. The surface of the whole figure is the sum of the areas of each of its sides.
It is convenient to study the surface using the example of a figure unfolding. A scan for a regular quadrangular pyramid is shown in the figures below.
We see that its surface area is equal to the sum of four areas of identical isosceles triangles and the area of a square.
The total area of all the triangles that form the sides of the figure is called the area of the lateral surface. Next, we show how to calculate it for a regular quadrangular pyramid.
Lateral surface area of a rectangular regular pyramid
To calculate the lateral surface area of the specified figure, we again turn to the above sweep. Suppose we know the side of the square base. Let's denote it by symbol a. It can be seen that each of the four identical triangles has a base of length a. To calculate their total area, you need to know this value for one triangle. From the course of geometry it is known that the area of \u200b\u200bthe triangle S t is equal to the product of the base and the height, which should be divided in half. I.e:
Where h b is the height of the isosceles triangle drawn to the base a. For a pyramid, this height is the apothem. Now it remains to multiply the resulting expression by 4 to get the area S b of the lateral surface for the pyramid in question:
S b = 4*S t = 2*h b *a.
This formula contains two parameters: the apothem and the side of the base. If the latter is known in most conditions of the problems, then the former has to be calculated knowing other quantities. Here are the formulas for calculating apotema h b for two cases:
- when the length of the side rib is known;
- when the height of the pyramid is known.
If we denote the length of the lateral edge (the side of an isosceles triangle) with the symbol L, then the apotema h b is determined by the formula:
h b \u003d √ (L 2 - a 2 / 4).
This expression is the result of applying the Pythagorean theorem for the lateral surface triangle.
If the height h of the pyramid is known, then the apotema h b can be calculated as follows:
h b = √(h 2 + a 2 /4).
It is also not difficult to obtain this expression if we consider a right triangle inside the pyramid formed by the legs h and a / 2 and the hypotenuse h b.
We will show how to apply these formulas by solving two interesting problems.
Problem with Known Surface Area
It is known that the area of the lateral surface of a quadrangular is 108 cm 2 . It is necessary to calculate the value of the length of its apothem h bif the height of the pyramid is 7 cm.
We write the formula for the area S b of the lateral surface through the height. We have:
S b = 2*√(h 2 + a 2 /4) *a.
Here we have simply substituted the corresponding apotema formula into the expression for S b . Let's square both sides of the equation:
S b 2 \u003d 4 * a 2 * h 2 + a 4.
To find the value of a, we make a change of variables:
t 2 + 4*h 2 *t - S b 2 = 0.
We now substitute the known values and solve the quadratic equation:
t 2 + 196*t - 11664 = 0.
We have written only the positive root of this equation. Then the sides of the base of the pyramid will be equal to:
a = √t = √47.8355 ≈ 6.916 cm.
To get the length of apotema, just use the formula:
h b \u003d √ (h 2 + a 2 / 4) \u003d √ (7 2 + 6.916 2 / 4) ≈ 7.808 cm.
Lateral surface of the pyramid of Cheops
Let's determine the value of the lateral surface area for the largest Egyptian pyramid. It is known that at its base lies a square with a side length of 230.363 meters. The height of the structure was originally 146.5 meters. Substitute these numbers into the corresponding formula for S b , we get:
S b \u003d 2 * √ (h 2 + a 2 / 4) * a \u003d 2 * √ (146.5 2 + 230.363 2 / 4) * 230.363 ≈ 85860 m 2.
The found value is slightly larger than the area of 17 football fields.
The surface area of the pyramid. In this article, we will consider with you problems with regular pyramids. Let me remind you that a regular pyramid is a pyramid whose base is a regular polygon, the top of the pyramid is projected into the center of this polygon.
The side face of such a pyramid is an isosceles triangle.The height of this triangle, drawn from the top of a regular pyramid, is called an apothem, SF is an apothem:
In the type of problems presented below, it is required to find the surface area of the entire pyramid or the area of its lateral surface. The blog has already considered several problems with regular pyramids, where the question was raised about finding elements (height, base edge, side edge), .
In the tasks of the exam, as a rule, regular triangular, quadrangular and hexagonal pyramids are considered. I have not seen problems with regular pentagonal and heptagonal pyramids.
The formula for the area of the entire surface is simple - you need to find the sum of the area of \u200b\u200bthe base of the pyramid and the area of its lateral surface:
Consider the tasks:
The sides of the base of a regular quadrangular pyramid are 72, the side edges are 164. Find the surface area of this pyramid.
The surface area of the pyramid is equal to the sum of the areas of the lateral surface and the base:
*The lateral surface consists of four triangles of equal area. The base of the pyramid is a square.
The area of the side of the pyramid can be calculated using:
Thus, the surface area of the pyramid is:
Answer: 28224
The sides of the base of a regular hexagonal pyramid are 22, the side edges are 61. Find the area of the lateral surface of this pyramid.
The base of a regular hexagonal pyramid is a regular hexagon.
The lateral surface area of this pyramid consists of six areas of equal triangles with sides 61.61 and 22:
Find the area of a triangle using Heron's formula:
So the lateral surface area is:
Answer: 3240
*In the problems presented above, the area of the side face could be found using a different triangle formula, but for this you need to calculate the apothem.
27155. Find the surface area of a regular quadrangular pyramid whose base sides are 6 and whose height is 4.
In order to find the surface area of a pyramid, we need to know the area of the base and the area of the side surface:
The area of the base is 36, since it is a square with a side of 6.
The side surface consists of four faces, which are equal triangles. In order to find the area of such a triangle, you need to know its base and height (apothem):
* The area of a triangle is equal to half the product of the base and the height drawn to this base.
The base is known, it is equal to six. Let's find the height. Consider a right triangle (highlighted in yellow):
One leg is equal to 4, since this is the height of the pyramid, the other is equal to 3, since it is equal to half the edge of the base. We can find the hypotenuse using the Pythagorean theorem:
So the area of the lateral surface of the pyramid is:
Thus, the surface area of the entire pyramid is:
Answer: 96
27069. The sides of the base of a regular quadrangular pyramid are 10, the side edges are 13. Find the surface area of this pyramid.
27070. The sides of the base of a regular hexagonal pyramid are 10, the side edges are 13. Find the area of the side surface of this pyramid.
There are also formulas for the lateral surface area of a regular pyramid. In a regular pyramid, the base is an orthogonal projection of the lateral surface, therefore:
P- perimeter of the base, l- apothem of the pyramid
*This formula is based on the formula for the area of a triangle.
If you want to learn more about how these formulas are derived, do not miss it, follow the publication of articles.That's all. Good luck to you!
Sincerely, Alexander Krutitskikh.
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- 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: 
