How to find the base area of a pyramid. How to calculate the area of a pyramid: base, lateral and full? What is a pyramid

28.10.2019

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 parties. 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 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 for 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 quadrilateral, 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 side surface area correct 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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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 triangle area formula, and you can also use the surface area formula 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 triangular pyramid area formula to calculate 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.

Task: let the correct one be given triangular pyramid. 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.

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's happened 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 everyone listed properties, making element calculations much easier. 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 a base equal angles.
When describing a circle around a polygon, all edges of the pyramid emanating from the vertex will have equal 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 right 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 school course geometries 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. Side surface includes only 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 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 area of the lateral surface of a regular quadrangular pyramid.

What is a pyramid?

We give a strict geometric definition pyramids. Suppose there is some polygon with n sides and n corners. Let's choose arbitrary point 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 ground is regular polygon, then the whole pyramid is called correct. 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 - height 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 inside the pyramid right triangle, formed by legs h and a/2 and 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.

Plug in the known values and solve 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 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 \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.

How to find the base area of ​​a pyramid. How to calculate the area of ​​a pyramid: base, lateral and full? What is a pyramid