Formula for the lateral surface of a regular quadrangular pyramid. Lateral surface area of different pyramids

The total area of the lateral surface of a pyramid consists of the sum of the areas of its lateral faces.

In a quadrangular pyramid, there are two types of faces - a quadrangle at the base and triangles with a common vertex, which form the side surface.
First you need to calculate the area of the side faces. To do this, you can use the formula for the area of a triangle, or you can also use the formula for the surface area of a quadrangular pyramid (only if the polyhedron is regular). If the pyramid is regular and the length of the edge a of the base and the apothem h drawn to it is known, then:

If, according to the conditions, the length of the edge c of a regular pyramid and the length of the side of the base a are given, then you can find the value using the following formula:

If the length of the edge at the base and the acute angle opposite it at the top are given, then the area of the lateral surface can be calculated by the ratio of the square of the side a to the double cosine of half the angle α:

Let's consider an example of calculating the surface area of a quadrangular pyramid through the side edge and the side of the base.

Problem: Let a regular quadrangular pyramid be given. Edge length b = 7 cm, base side length a = 4 cm. Substitute the given values into the formula:

We showed calculations of the area of one side face for a regular pyramid. Respectively. To find the area of the entire surface, you need to multiply the result by the number of faces, that is, by 4. If the pyramid is arbitrary and its faces are not equal to each other, then the area must be calculated for each individual side. If the base is a rectangle or parallelogram, then it is worth remembering their properties. The sides of these figures are parallel in pairs, and accordingly the faces of the pyramid will also be identical in pairs.
The formula for the area of the base of a quadrangular pyramid directly depends on which quadrilateral lies at the base. If the pyramid is correct, then the area of the base is calculated using the formula, if the base is a rhombus, then you will need to remember how it is located. If there is a rectangle at the base, then finding its area will be quite simple. It is enough to know the lengths of the sides of the base. Let's consider an example of calculating the area of the base of a quadrangular pyramid.

Problem: Let a pyramid be given, at the base of which lies a rectangle with sides a = 3 cm, b = 5 cm. An apothem is lowered from the top of the pyramid to each of the sides. h-a =4 cm, h-b =6 cm. The top of the pyramid lies on the same line as the point of intersection of the diagonals. Find the total area of the pyramid.
The formula for the area of a quadrangular pyramid consists of the sum of the areas of all faces and the area of the base. First, let's find the area of the base:

Now let's look at the sides of the pyramid. They are identical in pairs, because the height of the pyramid intersects the point of intersection of the diagonals. That is, in our pyramid there are two triangles with a base a and height h-a, as well as two triangles with a base b and height h-b. Now let's find the area of the triangle using the well-known formula:

Now let's perform an example of calculating the area of a quadrangular pyramid. In our pyramid with a rectangle at the base, the formula would look like this:

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.

Surface area of the pyramid. In this article we will look at 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 altitude of this triangle drawn from the vertex of a regular pyramid is called apothem, SF - apothem:

In the type of problem presented below, you need to find the surface area of the entire pyramid or the area of its lateral surface. The blog has already discussed several problems with regular pyramids, where the question was about finding the elements (height, base edge, side edge).

Unified State Examination tasks usually examine regular triangular, quadrangular and hexagonal pyramids. I haven’t seen any 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 the base of the pyramid and the area of its lateral surface:

Let's 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.

We can calculate the area of the side of the pyramid using:

Thus, the surface area of the pyramid is:

Answer: 28224

The sides of the base of a regular hexagonal pyramid are equal to 22, the side edges are equal to 61. Find the lateral surface area 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:

Let's find the area of the triangle using Heron's formula:

Thus, the lateral surface area is:

Answer: 3240

*In the problems presented above, the area of the side face could be found using another 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 the pyramid, we need to know the area of the base and the area of the lateral surface:

The area of the base is 36 since it is a square with side 6.

The lateral 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:

This means that 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 equal to 10, the side edges are equal to 13. Find the surface area of this pyramid.

27070. The sides of the base of a regular hexagonal pyramid are equal to 10, the side edges are equal to 13. Find the lateral surface area 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- base perimeter, 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, don’t miss it, follow the publication of articles.That's all. Good luck to you!

Sincerely, Alexander Krutitskikh.

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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.

What is a 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 of the entire figure is the sum of the areas of each of its sides.

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 = √(L 2 - a 2 /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:

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 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 area of the lateral surface of the quadrangular is 108 cm 2. 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*√(h 2 + a 2 /4) *a.

Here we simply substituted the appropriate apothem formula into the expression for S b. Let's square both sides of the equation:

S b 2 = 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.

Now we substitute the known values and solve the quadratic equation:

t 2 + 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 = √(h 2 + a 2 /4) = √(7 2 + 6.916 2 /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*√(h 2 + a 2 /4) *a = 2*√(146.5 2 +230.363 2 /4)*230.363 ≈ 85860 m 2.

The value found is slightly larger than the area of 17 football fields.

What figure do we call a pyramid? Firstly, 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 shape of triangles converging at one common vertex. Now, having understood the term, let’s find out how to find the surface area of the pyramid.

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

The choice of calculation formula depends on the shape of the polygon underlying our pyramid. It can be regular, that is, with sides of the same length, or irregular. Let's consider both options.

The base is a regular polygon

From the school course we know:

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 and multiplied by 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 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 .

At 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.

Lateral 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 lateral sides. There are also 2 options here.

Let us have an arbitrary pyramid, i.e. one with an irregular polygon at its base. Then you should calculate the area of each face separately and add the results. Since the sides of a pyramid, by definition, can only be triangles, the calculation is carried out using the above-mentioned formula: S=1/2a*h.
Let our pyramid be correct, i.e. at its base lies a regular polygon, and the projection of the top of the pyramid is at its center. Then, to calculate the area of the lateral surface (Sb), it is enough to find half the product of the perimeter of the base polygon (P) and the height (h) of the lateral side (the same for all faces): Sb = 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 algebraically calculate the surface areas of several pyramids.

Surface area of a triangular pyramid

At the base of such a pyramid is a triangle. Using the formula So=1/2a*h we find the area of the base. We use the same formula to find the area of each face of the pyramid, which also has 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 = S1+ S2+ S3. By adding up the areas of the sides and base, we obtain the total surface area of the desired pyramid: Sp= So+ Sb.

Surface area of a quadrangular pyramid

The area of the lateral surface is the sum of 4 terms: Sb = S1+ S2+ S3+ S4, each of which is calculated using the formula for the area of a triangle. And the area of the base will have to be looked for, depending on the shape of the quadrilateral - regular 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.