Surface area of a regular pyramid. Total surface area of the pyramid

What shape do we call a pyramid? First, 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 form of triangles converging at one common vertex. Now, having dealt with 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 the pyramid

The choice of the calculation formula depends on the shape of the polygon lying at the base of our pyramid. It can be correct, that is, with sides of the same length, or incorrect. Let's consider both options.

At the base is a regular polygon

From the school course it is known:

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 times 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 value of 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 .

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.

Side 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 sides. There are also 2 options here.

Let us have an arbitrary pyramid, i.e. one whose base is an irregular polygon. Then you should calculate separately the area of each face and add the results. Since the sides of the pyramid, by definition, can only be triangles, the calculation is based on the formula mentioned above: 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 in its center. Then, to calculate the area of the side surface (Sb), it is enough to find half the product of the perimeter of the base polygon (P) and the height (h) of the side (the same for all faces): Sb \u003d 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 calculate algebraically the surface areas of several pyramids.

Surface area of a triangular pyramid

At the base of such a pyramid is a triangle. According to the formula So \u003d 1 / 2a * h, we find the area of \u200b\u200bthe base. We apply the same formula to find the area of each face of the pyramid, also having 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 \u003d S1 + S2 + S3. Adding the areas of the sides and base, we get the total surface area of the desired pyramid: Sp \u003d So + Sb.

Surface area of a quadrangular pyramid

The lateral surface area is the sum of 4 terms: Sb \u003d S1 + S2 + S3 + S4, each of which is calculated using the triangle area formula. And the area of \u200b\u200bthe base will have to be sought, depending on the shape of the quadrangle - correct or irregular. The total surface area of the pyramid is again obtained by adding the base area and the total surface area of the given pyramid.

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

In a quadrangular pyramid, two types of faces are distinguished - a quadrilateral 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 formulas 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 drawn to it h is known in it, 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 rib at the base and the opposite acute angle at the apex are given, then the lateral surface area can be calculated by the ratio of the square of the side a to the doubled cosine of half the angle α:

Consider an example of calculating the surface area of a quadrangular pyramid through a side edge and a 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 have shown calculations of the area of one side face for a regular pyramid. Respectively. To find the area of the entire surface, it is necessary 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 it is necessary to calculate the area 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 \u200b\u200bthe base of a quadrangular pyramid directly depends on which quadrilateral lies at the base. If the pyramid is correct, then the area of \u200b\u200bthe base is calculated by the formula, if the base is a rhombus, then you need to remember how it is located. If the base is a rectangle, then finding its area will be quite simple. It is enough to know the lengths of the sides of the base. Consider an example of calculating the area of the base of a quadrangular pyramid.

Task: Let a pyramid be given, at the base of which lies a rectangle with sides a = 3 cm, b = 5 cm. An apothem is omitted from the top of the pyramid to each side. h-a \u003d 4 cm, h-b \u003d 6 cm. The top of the pyramid lies on the same line with the intersection point 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 consider the faces of the pyramid. They are identical in pairs, because the height of the pyramid intersects the intersection point of the diagonals. That is, in our pyramid there are two triangles with base a and height h-a, as well as two triangles with base b and height h-b. Now we 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 will look like this:

triangular pyramid A polyhedron is called a polyhedron whose base is a regular triangle.

In such a pyramid, the faces of the base and the edges of the sides are equal to each other. Accordingly, the area of the side faces is found from the sum of the areas of three identical triangles. You can find the lateral surface area of a regular pyramid using the formula. And you can make the calculation several times faster. To do this, apply the formula for the area of the lateral surface of a triangular pyramid:

where p is the perimeter of the base, all sides of which are equal to b, a is the apothem lowered from the top to this base. Consider an example of calculating the area of a triangular pyramid.

Task: Let the correct pyramid be given. The side of the triangle lying at the base is b = 4 cm. The apothem of the pyramid is a = 7 cm. Find the area of the lateral surface of the pyramid.
Since, according to the conditions of the problem, we know the lengths of all the necessary elements, we will find the perimeter. Remember that in a regular triangle, all sides are equal, and, therefore, the perimeter is calculated by the formula:

Substitute the data and find the value:

Now, knowing the perimeter, we can calculate the lateral surface area:

To apply the formula for the area of a triangular pyramid to calculate the full value, you need to find the area of the base of the polyhedron. For this, the formula is used:

The formula for the area of \u200b\u200bthe base of a triangular pyramid may be different. It is allowed to use any calculation of parameters for a given figure, but most often this is not required. Consider an example of calculating the area of the base of a triangular pyramid.

Task: In a regular pyramid, the side of the triangle lying at the base is a = 6 cm. Calculate the area of the base.
To calculate, we only need the length of the side of a regular triangle located at the base of the pyramid. Substitute the data in the formula:

Quite often it is required to find the total area of a polyhedron. To do this, you need to add the area of \u200b\u200bthe side surface and the base.

Consider an example of calculating the area of a triangular pyramid.

Problem: Let a regular triangular pyramid be given. The side of the base is b = 4 cm, the apothem is a = 6 cm. Find the total area of the pyramid.
First, let's find the lateral surface area using the already known formula. Calculate the perimeter:

We substitute the data in the formula:
Now find the area of the base:
Knowing the area of the base and lateral surface, we find the total area of \u200b\u200bthe pyramid:

When calculating the area of \u200b\u200ba regular pyramid, one should not forget that the base is a regular triangle and many elements of this polyhedron are equal to each other.

Instruction

First of all, it is worth understanding that the side 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 \u003d (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 \u003d (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 \u003d (a * b * c) / 4 * R, where R is the radius of the triangle described around the circle;

S \u003d (a * b) / 2 \u003d r² + 2 * r * R (if the triangle is right-angled);

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

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

Having calculated, using the above formulas, the areas of all triangles that are the faces of the pyramid, you can begin to calculate the area of \u200b\u200bthis 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 in a formula like this:

Sp = ΣSi, where Sp is the lateral area, 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: a regular pyramid is given, 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 of the lateral surface are 17 cm. Therefore, in order to calculate the area of \u200b\u200bany 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, we 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 the base, and the vertex is projected into the center of this polygon), then to calculate the entire side surface, it is enough to multiply the perimeter of the base (that is, the sum of the lengths of all sides of the polygon that lies at the base pyramid) by the height of the side face (otherwise called 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, then you will have to separately calculate the areas of all 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 are calculated, it remains only to add them up to get the area of the side surface of the pyramid.

Then you need to calculate the area of \u200b\u200bthe base of the pyramid. The choice of the formula for the calculation depends on which polygon lies at the base of the pyramid: correct (that is, one whose all sides have the same length) or incorrect. The area of a regular polygon can be calculated by multiplying the perimeter by the radius of the circle inscribed 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 circle inscribed in the polygon .

A truncated pyramid is a polyhedron formed by a pyramid and its section parallel to the base. Finding the area of the lateral surface 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. Consider an example of calculating the lateral surface area. Let's say a regular pyramid is given. 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 a large base, it will be equal to p1=4b=4*5=20 cm. In a 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.

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.