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 height 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 of finding elements (height, base edge, side edge) was raised.

IN Unified State Exam assignments As a rule, regular triangular, quadrangular and hexagonal pyramids are considered. 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. Let's consider right triangle(it's 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!

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Pyramid- one of the varieties of a polyhedron formed from polygons and triangles that lie at the base and are its faces.

Moreover, at the top of the pyramid (i.e. at one point) all the faces are united.

In order to calculate the area of ​​a pyramid, it is worth determining that its lateral surface consists of several triangles. And we can easily find their areas using

various formulas. Depending on what data we know about the triangles, we look for their area.

We list some formulas that can be used to find the area of ​​triangles:

1. S = (a*h)/2 . IN in this case we know the height of the triangle h , which is lowered to the side a .
2. S = a*b*sinβ . Here are the sides of the triangle a , b , and the angle between them is β .
3. S = (r*(a + b + c))/2 . Here are the sides of the triangle a, b, c . The radius of a circle that is inscribed in a triangle is r .
4. S = (a*b*c)/4*R . The radius of a circumscribed circle around a triangle is R .
5. S = (a*b)/2 = r² + 2*r*R . This formula needs to be applied only when the triangle is right-angled.
6. S = (a²*√3)/4 . We apply this formula to an equilateral triangle.

Only after we calculate the areas of all the triangles that are the faces of our pyramid can we calculate the area of ​​its lateral surface. To do this, we will use the above formulas.

In order to calculate the area of ​​the lateral surface of a pyramid, no difficulties arise: you need to find out the sum of the areas of all triangles. Let's express this with the formula:

Sp = ΣSi

Here Si is the area of ​​the first triangle, and S P - area of ​​the lateral surface of the pyramid.

Let's look at an example. Given a regular pyramid, its lateral faces are formed by several equilateral triangles,

« Geometry is the most powerful tool for sharpening our mental abilities».

Galileo Galilei.

and the square is the base of the pyramid. Moreover, the edge of the pyramid has a length of 17 cm. Let us find the area of ​​the lateral surface of this pyramid.

We reason like this: we know that the faces of the pyramid are triangles, they are equilateral. We also know what the edge length of this pyramid is. It follows that all triangles have equal sides and their length is 17 cm.

To calculate the area of ​​each of these triangles, you can use the following formula:

S = (17²*√3)/4 = (289*1.732)/4 = 125.137 cm²

So, since we know that a square lies at the base of the pyramid, it turns out that we have four equilateral triangles. This means that the lateral surface area of ​​the pyramid can be easily calculated using the following formula: 125.137 cm² * 4 = 500.548 cm²

Our answer is as follows: 500.548 cm² - this is the area of ​​the lateral surface of this pyramid.

Instructions

First of all, it is worth understanding that the lateral surface of the pyramid is represented by several triangles, the areas of which can be found using a variety of formulas, depending on the known data:

S = (a*h)/2, where h is the height lowered to side a;

S = a*b*sinβ, where a, b are the sides of the triangle, and β is the angle between these sides;

S = (r*(a + b + c))/2, where a, b, c are the sides of the triangle, and r is the radius of the circle inscribed in this triangle;

S = (a*b*c)/4*R, where R is the radius of the triangle circumscribed around the circle;

S = (a*b)/2 = r² + 2*r*R (if the triangle is right-angled);

S = S = (a²*√3)/4 (if the triangle is equilateral).

In fact, these are only the most basic known formulas for finding the area of ​​a triangle.

Having calculated the areas of all triangles that are the faces of the pyramid using the above formulas, you can begin to calculate the area of ​​this pyramid. This is done extremely simply: you need to add up the areas of all the triangles that form the side surface of the pyramid. This can be expressed by the formula:

Sp = ΣSi, where Sp is the area of ​​the lateral surface, Si is the area of ​​the i-th triangle, which is part of its lateral surface.

For greater clarity, we can consider a small example: given a regular pyramid, the side faces of which are formed by equilateral triangles, and at its base lies a square. The length of the edge of this pyramid is 17 cm. It is required to find the area of ​​the lateral surface of this pyramid.

Solution: the length of the edge of this pyramid is known, it is known that its faces are equilateral triangles. Thus, we can say that all sides of all triangles on the lateral surface are equal to 17 cm. Therefore, in order to calculate the area of ​​​​any of these triangles, you will need to apply the formula:

S = (17²*√3)/4 = (289*1.732)/4 = 125.137 cm²

It is known that at the base of the pyramid lies a square. Thus, it is clear that there are four given equilateral triangles. Then the area of ​​the lateral surface of the pyramid is calculated as follows:

125.137 cm² * 4 = 500.548 cm²

Answer: The lateral surface area of ​​the pyramid is 500.548 cm²

First, let's calculate the area of ​​the lateral surface of the pyramid. The lateral surface is the sum of the areas of all lateral faces. If you are dealing with a regular pyramid (that is, one that has a regular polygon at its base, and the vertex is projected into the center of this polygon), then to calculate the entire lateral surface it is enough to multiply the perimeter of the base (that is, the sum of the lengths of all sides of the polygon lying at the base pyramid) by the height of the side face (otherwise called the apothem) and divide the resulting value by 2: Sb = 1/2P*h, where Sb is the area of ​​the side surface, P is the perimeter of the base, h is the height of the side face (apothem).

If you have an arbitrary pyramid in front of you, you will have to separately calculate the areas of all the faces and then add them up. Since the side faces of the pyramid are triangles, use the formula for the area of ​​a triangle: S=1/2b*h, where b is the base of the triangle, and h is the height. When the areas of all the faces have been calculated, all that remains is to add them up to get the area of ​​the lateral surface of the pyramid.

Then you need to calculate the area of ​​the base of the pyramid. The choice of formula for calculation depends on which polygon lies at the base of the pyramid: regular (that is, one with all sides of the same length) or irregular. The area of ​​a regular polygon can be calculated by multiplying the perimeter by the radius of the inscribed circle in the polygon and dividing the resulting value by 2: Sn = 1/2P*r, where Sn is the area of ​​the polygon, P is the perimeter, and r is the radius of the inscribed circle in the polygon .

A truncated pyramid is a polyhedron that is formed by a pyramid and its cross section parallel to the base. Finding the lateral surface area of ​​the pyramid is not difficult at all. Its very simple: the area is equal to the product of half the sum of the bases by . Let's consider an example of calculating the lateral surface area. Suppose we are given a regular pyramid. The lengths of the base are b = 5 cm, c = 3 cm. Apothem a = 4 cm. To find the area of ​​the lateral surface of the pyramid, you must first find the perimeter of the bases. IN large base it will be equal to p1=4b=4*5=20 cm. In the smaller base the formula will be as follows: p2=4c=4*3=12 cm. Therefore, the area will be equal to: s=1/2(20+12)*4 =32/2*4=64 cm.

If there is an irregular polygon at the base of the pyramid, to calculate the area of ​​the entire figure, you will first need to break the polygon into triangles, calculate the area of ​​each, and then add them. In other cases, to find the side surface of the pyramid, you need to find the area of ​​​​each of its side faces and add up the results. In some cases, the task of finding the side surface of the pyramid can be made easier. If one side face is perpendicular to the base or two adjacent side faces are perpendicular to the base, then the base of the pyramid is considered an orthogonal projection of part of its side surface, and they are related by formulas.

To complete the calculation of the surface area of ​​the pyramid, add the areas of the side surface and the base of the pyramid.

A pyramid is a polyhedron, one of whose faces (base) is an arbitrary polygon, and the remaining faces (sides) are triangles having . According to the number of angles, the bases of the pyramid are triangular (tetrahedron), quadrangular, and so on.

A pyramid is a polyhedron with a base in the form of a polygon, and the remaining faces are triangles with a common vertex. An apothem is the height of the side face of a regular pyramid, which is drawn from its vertex.

A pyramid is a polyhedron, the base of which is a polygon, and the side faces are triangles that have one common vertex. Square surfaces pyramids equal to the sum of the areas of the lateral surfaces and grounds pyramids.

You will need

• Paper, pen, calculator

Instructions

First we calculate the area of ​​the side surfaces . By lateral surface we mean the sum of all lateral faces. If you are dealing with a regular pyramid (that is, one in which a regular polygon lies, and the vertex is projected to the center of this polygon), then to calculate the entire lateral surfaces it is enough to multiply the perimeter of the base (that is, the sum of the lengths of all sides of the polygon lying at the base pyramids) by the height of the side face (otherwise called) and divide the resulting value by 2: Sb=1/2P*h, where Sb is the area of ​​the side surfaces, P - perimeter of the base, h - height of the side face (apothem).

If you have an arbitrary pyramid in front of you, you will have to calculate the areas of all the faces and then add them up. Since the side faces pyramids are, use the formula for the area of ​​a triangle: S=1/2b*h, where b is the base of the triangle, and h is the height. When the areas of all the faces have been calculated, all that remains is to add them up to get the area of ​​the side surfaces pyramids.

Then you need to calculate the area of ​​the base pyramids. The choice for calculation depends on whether the polygon lies at the base of the pyramid: regular (that is, one whose sides are all the same length) or. Square of a regular polygon can be calculated by multiplying the perimeter by the radius of the inscribed circle in the polygon and dividing the resulting value by 2: Sn = 1/2P*r, where Sn is the area of ​​the polygon, P is the perimeter, and r is the radius of the inscribed circle in the polygon.

If at the base pyramids lies an irregular polygon, then to calculate the area of ​​the entire figure you will again have to divide the polygon into triangles, calculate the area of ​​each, and then add them.

To complete the area calculation surfaces pyramids, fold the square side surfaces and grounds pyramids.

Video on the topic

The polygon represents geometric figure, constructed by closing a broken line. There are several types of polygon, which differ depending on the number of vertices. The area is calculated for each type of polygon in certain ways.

Instructions

Multiply the lengths of the sides if you need to calculate the area of ​​a square or rectangle. If you need to find out the area of ​​a right triangle, extend it to a rectangle, calculate its area and divide it by two.

Use the following method to calculate the area if the figure does not have more than 180 degrees (a convex polygon), while all its vertices are in the coordinate grid, and does not intersect itself.
Draw a rectangle around such a polygon so that its sides are parallel to the grid lines (coordinate axes). In this case, at least one of the vertices of the polygon must be the vertex of a rectangle.

Only a truncated one can have two bases pyramids. In this case, the second base is formed by a section parallel to the larger base pyramids. Find one of reasons possible if it is known or linear elements of the second.

You will need

• - properties of the pyramid;
• - trigonometric functions;
• - similarity of figures;
• - finding the areas of polygons.

Instructions

If the base is a regular triangle, find it square by multiplying the square of the side by the square root of 3 divided by 4. If the base is a square, raise its side to the second power. In general, for any regular polygon, apply the formula S=(n/4) a² ctg(180º/n), where n is the number of sides of the regular polygon, a is the length of its side.

Find the side of the smaller base using the formula b=2 (a/(2 tg(180º/n))-h/tg(α)) tg(180º/n). Here a is the larger base, h is the height of the truncated pyramids, α – dihedral angle at its base, n – number of sides reasons(it's the same). Find the area of ​​the second base similarly to the first, using in the formula the length of its side S=(n/4) b² ctg(180º/n).

If the bases are other types of polygons, all sides of one of them are known reasons, and one of the sides of the other, then calculate the remaining sides as similar. For example, the sides of the larger base are 4, 6, 8 cm. The larger side of the smaller base is 4 cm. Calculate the proportionality coefficient, 4/8 = 2 (we take the sides in each of reasons), and calculate the other sides 6/2=3 cm, 4/2=2 cm. We get sides 2, 3, 4 cm at the smaller base of the side. Now calculate them as the areas of the triangles.

If the ratio of the corresponding elements in the truncated one is known, then the ratio of the areas reasons will be equal to the ratio of the squares of these elements. For example, if the relevant parties are known reasons a and a1, then a²/a1²=S/S1.

Under area pyramids usually refers to the area of ​​its lateral or total surface. Based on this geometric body lies a polygon. The side edges are triangular in shape. They have a common vertex, which is also the vertex pyramids.

You will need

• - paper;
• - pen;
• - calculator;
• - a pyramid with given parameters.

Instructions

Consider the pyramid given in the task. Determine whether the polygon is regular or irregular at its base. The correct one has all sides equal. The area in this case is equal to half the product of the perimeter and the radius. Find the perimeter by multiplying the length of the side l by the number of sides n, that is, P=l*n. The area of ​​the base can be expressed by the formula So=1/2P*r, where P is the perimeter, and r is the radius of the inscribed circle.

The perimeter and area of ​​an irregular polygon are calculated differently. The parties have different lengths. To

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 for 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 the legs h and a/2 and the 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.

Let's substitute now known values and decide 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.

Briefly about the main thing

Surface area (2019)

Prism surface area

Is there a general formula? No, in general, no. You just need to look for the areas of the side faces and sum them up.

The formula can be written for straight prism:

Where is the perimeter of the base.

But it’s still much easier to add up all the areas in each specific case than to memorize additional formulas. For example, let's calculate the total surface of a regular hexagonal prism.

All side faces are rectangles. Means.

This was already shown when calculating the volume.

So we get:

Surface area of ​​the pyramid

The general rule also applies to the pyramid:

Now let's calculate the surface area of ​​the most popular pyramids.

Surface area of ​​a regular triangular pyramid

Let the side of the base be equal and side rib equals. We need to find and.

Let us now remember that

This is the area of ​​a regular triangle.

And let’s remember how to look for this area. We use the area formula:

For us, “ ” is this, and “ ” is also this, eh.

Now let's find it.

Using the basic area formula and the Pythagorean theorem, we find

Attention: if you have a regular tetrahedron (i.e.), then the formula turns out like this:

Surface area of ​​a regular quadrangular pyramid

Let the side of the base be equal and the side edge equal.

The base is a square, and that's why.

It remains to find the area of ​​the side face

Surface area of ​​a regular hexagonal pyramid.

Let the side of the base be equal and the side edge.

How to find? A hexagon consists of exactly six identical regular triangles. We have already looked for the area of ​​a regular triangle when calculating the surface area of ​​a regular triangular pyramid; here we use the formula we found.

Well, we’ve already looked for the area of ​​the side face twice.

Well, the topic is over. If you are reading these lines, it means you are very cool.

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