is a multifaceted figure, the base of which is a polygon, and the remaining faces are represented by triangles with a common vertex.
If the base is a square, then the pyramid is called quadrangular, if a triangle – then triangular. The height of the pyramid is drawn from its top perpendicular to the base. Also used to calculate area apothem– the height of the side face, lowered from its top.
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 the pyramid is calculated through the perimeter of the base and the apothem:
Let's consider an example of calculating the area of the lateral surface of a pyramid.
Let a pyramid be given with base ABCDE and top F. 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 edges of the base are equal, the perimeter of the pentagon will be equal to:
Now you can find the lateral area of the pyramid: 
Area of a regular triangular pyramid
A regular triangular pyramid consists of a base in which lies a regular triangle 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 different ways. You can apply the usual calculation formula using the perimeter and apothem, or you can find the area of one face and multiply it by three. Since the face of a 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. Let's consider an example of calculating the lateral surface area of a regular triangular pyramid.
Given a pyramid with apothem a = 4 cm and 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 into the formula: 
Since in a regular pyramid all the 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: 
Area of a truncated pyramid
Truncated A pyramid is a polyhedron that is formed by a pyramid and its cross 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: 
Before studying questions about this geometric figure and its properties, you should understand some terms. When a person hears about a pyramid, he imagines huge buildings in Egypt. This is what the simplest ones look like. But they happen different types and shapes, which means the calculation formula for geometric shapes will be different.
Types of figure
Pyramid – geometric figure , denoting and representing several faces. In essence, 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 comes in two main types:
- correct;
- truncated.
In the first case, the base is a regular polygon. Here all lateral 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 cross section formed parallel to the base.
Terms and symbols
Key terms:
- Regular (equilateral) triangle- a figure with three equal angles and equal sides. In this case, all angles are 60 degrees. The figure is the simplest of regular polyhedra. If this figure lies at the base, then such a polyhedron will be called regular triangular. 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 apex is formed by a straight line extending from the apex to the base of the pyramid.
- Edge– 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.
- Section- a flat figure formed as a result of dissection. It should not be confused with a section, since 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 located. This definition is valid only in relation to a regular polyhedron. For example, if this is not a truncated pyramid, then the face will be a triangle. In this case, the height of this triangle will become the apothem.
Area formulas
Find the lateral surface area 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 each face and add them together.
Depending on what parameters are known, formulas for calculating a square, trapezoid, arbitrary quadrilateral, etc. may be required. The formulas themselves in different cases will also have differences.
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 specifically for such figures. Therefore, the corresponding formulas will be given below. Otherwise, you would have to write everything out over several pages, which would only confuse and confuse you.
Basic formula for calculation The lateral surface area of a regular pyramid will have the following form:
S=½ Pa (P is the perimeter of the base, and is the apothem)
Let's look at one example. The polyhedron has a base with segments A1, A2, A3, A4, A5, and all of them are 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, you can find it like this: P = 5 * 10 = 50 cm. Next, we apply the basic formula: S = ½ * 50 * 5 = 125 cm squared.
Lateral surface area of a regular triangular pyramid easiest to calculate. The formula looks like this:
S =½* ab *3, where a is the apothem, b is the face 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. Let's look at an example. Given a figure with an apothem of 5 cm and a base edge 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 =1/2*(p_01+ p_02)*a, where p_01 and p_02 are the perimeters of the bases, and is the apothem. Let's look at an example. Let’s say that for a quadrangular figure the dimensions of the sides of the bases are 3 and 6 cm, and the apothem is 4 cm.
Here, first you need to find the perimeters of the bases: р_01 =3*4=12 cm; р_02=6*4=24 cm. It remains to substitute the values into the main formula and we get: S =1/2*(12+24)*4=0.5*36*4=72 cm squared.
Thus, you can find the lateral surface area of a regular pyramid of any complexity. You should be careful and not confuse these calculations with the total area of the entire polyhedron. And if you still need to do this, just calculate the area of the largest base of the polyhedron and add it to the area of the lateral surface of the polyhedron.
Video
This video will help you consolidate information on how to find the lateral surface area of different pyramids.
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Typical geometric problems on 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 lateral surface area of a regular quadrangular pyramid.
Let us give a strict geometric definition of a pyramid. Suppose we have a polygon with n sides and n angles. Let's 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 with a certain volume, which is called an n-gonal pyramid. For example, let's show in the figure below what a pentagonal pyramid looks like.
The two important elements of any pyramid are its base (n-gon) and its apex. These elements are connected to each other by n triangles, which in general are not equal to each other. The perpendicular descending from the top to the base is called the height of the figure. If it intersects the base at 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 picture below shows what regular pyramids look like with triangular, quadrangular, pentagonal and hexagonal bases.
Surface of the pyramid
Before moving on to the question of the lateral surface area of a regular quadrangular pyramid, we 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 area of the entire figure is the sum of the areas of each side.
It is convenient to study a surface using the example of the development of a figure. The development 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 triangles that form the sides of a figure is usually called the lateral surface area. Next we will show how to calculate it for a regular quadrangular pyramid.
Lateral surface area of a quadrangular regular pyramid
To calculate the lateral surface area of the indicated figure, we again turn to the above development. Let's assume that we know the side of the square base. Let's denote it by the 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 geometry course we know that the area S t of a triangle is equal to the product of the base and the height, which should be divided in half. That is:
Where h b is the height of an isosceles triangle drawn to the base a. For a pyramid, this height is an apothem. Now it remains to multiply the resulting expression by 4 to obtain 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 problem conditions, then the former has to be calculated knowing other quantities. Here are the formulas for calculating the apothem 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 (side of an isosceles triangle) by the symbol L, then the apothem h b is determined by the formula:
h b = √(L2 - a2/4).
This expression is the result of applying the Pythagorean theorem to the lateral surface triangle.
If the height h of the pyramid is known, then the apothem h b can be calculated as follows:
It is also not difficult to obtain this expression if we consider a right triangle inside the pyramid, formed by legs h and a/2 and hypotenuse h b.
Let's show how to apply these formulas by solving two interesting problems.
Problem with known surface area
It is known that the lateral surface area of a regular quadrangular pyramid is 108 cm2. It is necessary to calculate the length of its apothem h b if the height of the pyramid is 7 cm.
Let us write the formula for the area S b of the lateral surface in terms of height. We have:
S b = 2*√(h2 + a2/4) *a.
Here we simply substituted the appropriate apothem formula into the expression for S b. Let's square both sides of the equation:
To find the value of a, we make a change of variables:
t2 + 4*h2*t - S b 2 = 0.
Now we substitute the known values and solve the quadratic equation:
t2 + 196*t – 11664 = 0.
We have written down 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 the apothem, just use the formula:
h b = √(h2 + a2/4) = √(72 + 6.9162/4) ≈ 7.808 cm.
Side surface of the Cheops pyramid
Let us 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 = 2*√(h2 + a2/4) *a = 2*√(146.52+230.3632/4)*230.363 ≈ 85860 m2.
The value found is slightly larger than the area of 17 football fields.
Lateral surface area of a regular quadrangular pyramid: formulas and example problems - all about traveling to the site
When preparing for the Unified State Exam in mathematics, students have to systematize their knowledge of algebra and geometry. I would like to combine all known information, for example, on how to calculate the area of a pyramid. Moreover, starting from the base and side edges to the entire surface area. If the situation with the side faces is clear, since they are triangles, then the base is always different.
How to find the area of the base of the pyramid?
It can be absolutely any figure: from an arbitrary triangle to an n-gon. And this base, in addition to the difference in the number of angles, can be a regular figure or an irregular one. In the Unified State Exam tasks that interest schoolchildren, there are only tasks with correct figures at the base. Therefore, we will talk only about them.
Regular triangle
That is, equilateral. The one in which all sides are equal and are designated by the letter “a”. In this case, the area of the base of the pyramid is calculated by the formula:
S = (a 2 * √3) / 4.
Square
The formula for calculating its area is the simplest, here “a” is again the side:
Arbitrary regular n-gon
The side of a polygon has the same notation. For the number of angles, the Latin letter n is used.
S = (n * a 2) / (4 * tg (180º/n)).
What to do when calculating the lateral and total surface area?
Since the base is a regular figure, all faces of the pyramid are equal. Moreover, each of them is an isosceles triangle, since the side edges are equal. Then, in order to calculate the lateral area of the pyramid, you will need a formula consisting of the sum of identical monomials. The number of terms is determined by the number of sides of the base.
The area of an isosceles triangle is calculated by the formula in which half the product of the base is multiplied by the height. This height in the pyramid is called apothem. Its designation is “A”. The general formula for lateral surface area is:
S = ½ P*A, where P is the perimeter of the base of the pyramid.
There are situations when the sides of the base are not known, but the side edges (c) and the flat angle at its apex (α) are given. Then you need to use the following formula to calculate the lateral area of the pyramid:
S = n/2 * in 2 sin α .
Task No. 1
Condition. Find the total area of the pyramid if its base has a side of 4 cm and the apothem has a value of √3 cm.
Solution. You need to start by calculating the perimeter of the base. Since this is a regular triangle, then P = 3*4 = 12 cm. Since the apothem is known, we can immediately calculate the area of the entire lateral surface: ½*12*√3 = 6√3 cm 2.
For the triangle at the base, you get the following area value: (4 2 *√3) / 4 = 4√3 cm 2.
To determine the entire area, you will need to add the two resulting values: 6√3 + 4√3 = 10√3 cm 2.
Answer. 10√3 cm 2.
Problem No. 2
Condition. There is a regular quadrangular pyramid. The length of the base side is 7 mm, the side edge is 16 mm. It is necessary to find out its surface area.
Solution. Since the polyhedron is quadrangular and regular, its base is a square. Once you know the area of the base and side faces, you will be able to calculate the area of the pyramid. The formula for the square is given above. And for the side faces, all sides of the triangle are known. Therefore, you can use Heron's formula to calculate their areas.
The first calculations are simple and lead to the following number: 49 mm 2. For the second value, you will need to calculate the semi-perimeter: (7 + 16*2): 2 = 19.5 mm. Now you can calculate the area of an isosceles triangle: √(19.5*(19.5-7)*(19.5-16) 2) = √2985.9375 = 54.644 mm 2. There are only four such triangles, so when calculating the final number you will need to multiply it by 4.
It turns out: 49 + 4 * 54.644 = 267.576 mm 2.
Answer. The desired value is 267.576 mm 2.
Problem No. 3
Condition. For a regular quadrangular pyramid, you need to calculate the area. The side of the square is known to be 6 cm and the height is 4 cm.
Solution. The easiest way is to use the formula with the product of perimeter and apothem. The first value is easy to find. The second one is a little more complicated.
We will have to remember the Pythagorean theorem and consider It is formed by the height of the pyramid and the apothem, which is the hypotenuse. The second leg is equal to half the side of the square, since the height of the polyhedron falls into its middle.
The required apothem (hypotenuse of a right triangle) is equal to √(3 2 + 4 2) = 5 (cm).
Now you can calculate the required value: ½*(4*6)*5+6 2 = 96 (cm 2).
Answer. 96 cm 2.
Problem No. 4
Condition. The correct side is given. The sides of its base are 22 mm, the side edges are 61 mm. What is the lateral surface area of this polyhedron?
Solution. The reasoning in it is the same as that described in task No. 2. Only there was given a pyramid with a square at the base, and now it is a hexagon.
First of all, the base area is calculated using the above formula: (6*22 2) / (4*tg (180º/6)) = 726/(tg30º) = 726√3 cm 2.
Now you need to find out the semi-perimeter of an isosceles triangle, which is the side face. (22+61*2):2 = 72 cm. All that remains is to use Heron’s formula to calculate the area of each such triangle, and then multiply it by six and add it to the one obtained for the base.
Calculations using Heron's formula: √(72*(72-22)*(72-61) 2)=√435600=660 cm 2. Calculations that will give the lateral surface area: 660 * 6 = 3960 cm 2. It remains to add them up to find out the entire surface: 5217.47≈5217 cm 2.
Answer. The base is 726√3 cm2, the side surface is 3960 cm2, the entire area is 5217 cm2.
