IN school course In stereometry, one of the simplest figures, which has non-zero dimensions along three spatial axes, is a quadrangular prism. Let's consider in the article what kind of figure this is, what elements it consists of, and also how you can calculate its surface area and volume.
The concept of a prism
In geometry, a prism is a spatial figure formed by two on the same grounds and lateral surfaces that connect the sides of these bases. Note that both bases are transferred to each other using the operation of parallel transfer to a certain vector. This definition of a prism leads to the fact that all its sides are always parallelograms.
The number of sides of the base can be arbitrary, starting from three. As this number tends to infinity, the prism smoothly turns into a cylinder, since its base becomes a circle, and the side parallelograms, connecting, form a cylindrical surface.
Like any polyhedron, a prism is characterized by sides (planes that limit the figure), edges (segments along which any two sides intersect) and vertices (meeting points of three sides, for a prism two of them are lateral, and the third is the base). The quantities of the three named elements of the figure are related to each other by the following expression:
Here P, C and B are the number of edges, sides and vertices, respectively. This expression is a mathematical representation of Euler's theorem.
Above is a picture showing two prisms. At the base of one of them (A) lies a regular hexagon, and the lateral sides are perpendicular to the bases. Figure B shows another prism. Its sides are no longer perpendicular to the bases, and the base is regular pentagon.
quadrangular?
As is clear from the description above, the type of prism is primarily determined by the type of polygon that forms the base (both bases are the same, so we can talk about one of them). If this polygon is a parallelogram, then we get a quadrangular prism. So all the sides of this are parallelograms. A quadrangular prism has its own name - parallelepiped.
The number of sides of a parallelepiped is six, with each side having a parallel parallelepiped similar to it. Since the bases of the parallelepiped are two sides, the remaining four are lateral.
The number of vertices of a parallelepiped is eight, which is easy to see if you remember that the vertices of a prism are formed only at the vertices of the base polygons (4x2=8). Applying Euler's theorem, we obtain the number of edges:
P = C + B - 2 = 6 + 8 - 2 = 12
Of the 12 ribs, only 4 are formed independently by the lateral sides. The remaining 8 lie in the planes of the figure’s bases.
Types of parallelepipeds
The first type of classification lies in the features of the parallelogram lying at the base. It may look like this:
- ordinary, whose angles are not equal to 90 o;
- rectangle;
- a square is a regular quadrilateral.
The second type of classification is the angle at which the side intersects the base. There are two different cases possible here:
- this angle is not right, then the prism is called oblique or inclined;
- the angle is 90 o, then such a prism is rectangular or simply straight.
The third type of classification is related to the height of the prism. If the prism is rectangular and has either a square or a rectangle at its base, then it is called a cuboid. If there is a square at the base, the prism is rectangular, and its height is equal to the length of the side of the square, then we get the well-known figure of a cube.
Prism surface and area
The set of all points that lie on the two bases of the prism (parallelograms) and on its sides (four parallelograms) form the surface of the figure. The area of this surface can be calculated by calculating the area of the base and this value for the side surface. Then their sum will give the desired value. Mathematically it is written like this:
Here S o and S b are the area of the base and lateral surface, respectively. The number 2 before S o appears because there are two bases.
Note that the written formula is valid for any prism, and not just for the area of a quadrangular prism.
It is useful to recall that the area of a parallelogram S p is calculated by the formula:
Where the symbols a and h denote the length of one of its sides and the height drawn to this side, respectively.
Area of a rectangular prism with a square base
The base is a square. For definiteness, let us denote its side by the letter a. To calculate the area of a regular quadrangular prism, you need to know its height. According to the definition for this value, it is equal to the length of the perpendicular dropped from one base to another, that is, equal to the distance between them. Let's denote it by the letter h. Since all the side faces are perpendicular to the bases for the type of prism under consideration, the height of a regular quadrangular prism will be equal to the length of its side edge.
The general formula for the surface area of a prism has two terms. The area of the base in this case is easy to calculate, it is equal to:
To calculate the area of the lateral surface, we reason as follows: this surface is formed by 4 identical rectangles. Moreover, the sides of each of them are equal to a and h. This means that the area S b will be equal to:
Note that the product 4*a is the perimeter of the square base. If we generalize this expression to the case of an arbitrary base, then for a rectangular prism the lateral surface can be calculated as follows:
Where P o is the perimeter of the base.
Returning to the problem of calculating the area of a regular quadrangular prism, we can write the final formula:
S = 2*S o + S b = 2*a 2 + 4*a*h = 2*a*(a+2*h)
Area of an oblique parallelepiped
It is somewhat more difficult to calculate than for a rectangular one. In this case, the area of the base of a quadrangular prism is calculated using the same formula as for a parallelogram. The changes concern the method for determining the lateral surface area.
To do this, use the same formula through the perimeter as given in the paragraph above. Only now it will have slightly different multipliers. The general formula for S b in the case of an oblique prism is:
Here c is the length of the side edge of the figure. The value P sr is the perimeter of the rectangular cut. This environment is constructed as follows: it is necessary to intersect all the side faces with a plane so that it is perpendicular to all of them. The resulting rectangle will be the desired cut.
The figure above shows an example of an oblique parallelepiped. Its shaded section with the sides forms right angles. The perimeter of the section is P sr. It is formed by four heights of lateral parallelograms. For this quadrangular prism, the lateral surface area is calculated using the above formula.
Diagonal length of a rectangular parallelepiped
The diagonal of a parallelepiped is a segment that connects two vertices that do not have common sides that form them. Any quadrangular prism has only four diagonals. For a rectangular parallelepiped with a rectangle at its base, the lengths of all diagonals are equal to each other.
The figure below shows the corresponding figure. The red segment is its diagonal.
D = √(A 2 + B 2 + C 2)
Here D is the length of the diagonal. The remaining symbols are the lengths of the sides of the parallelepiped.
Many people confuse the diagonal of a parallelepiped with the diagonals of its sides. Below is a drawing where the diagonals of the sides of the figure are depicted in colored segments.
The length of each of them is also determined by the Pythagorean theorem and is equal to square root from the sum of the squares of the corresponding lengths of the sides.
Prism volume
In addition to the area of a regular quadrangular prism or other types of prisms, to solve some geometric problems you should also know their volume. This value for absolutely any prism is calculated using the following formula:
If the prism is rectangular, then it is enough to calculate the area of its base and multiply it by the length of the side edge to obtain the volume of the figure.
If the prism is regular quadrangular, then its volume will be equal to:
It is easy to see that this formula transforms into an expression for the volume of a cube if the length of the side edge h is equal to the side of the base a.
Problem with a rectangular parallelepiped
To consolidate the studied material, we will solve the following problem: there is a rectangular parallelepiped, the sides of which are 3 cm, 4 cm and 5 cm. It is necessary to calculate its surface area, diagonal length and volume.
S = 2*S o + S b = 2*12 + 5*14 = 24 + 70 = 94 cm 2
To determine the length of the diagonal and the volume of the figure, you can directly use the above expressions:
D = √(3 2 +4 2 +5 2) = 7.071 cm;
V = 3*4*5 = 60 cm3.
Oblique parallelepiped problem
The figure below shows an oblique prism. Its sides are equal: a = 10 cm, b = 8 cm, c = 12 cm. It is necessary to find the surface area of this figure.
First, let's determine the area of the base. From the figure it is clear that acute angle equal to 50 o. Then its area is equal to:
S o = h*a = sin(50 o)*b*a
To determine the lateral surface area, find the perimeter of the shaded rectangle. The sides of this rectangle are a*sin(45 o) and b*sin(60 o). Then the perimeter of this rectangle is:
P sr = 2*(a*sin(45 o)+b*sin(60 o))
The total surface area of this parallelepiped is:
S = 2*S o + S b = 2*(sin(50 o)*b*a + a*c*sin(45 o) + b*c*sin(60 o))
We substitute the data from the problem conditions for the lengths of the sides of the figure, and we get the answer:
From the solution to this problem it is clear that trigonometric functions are used to determine the areas of oblique figures.
IN school curriculum In a stereometry course, the study of three-dimensional figures usually begins with a simple geometric body - the polyhedron of a prism. The role of its bases is performed by 2 equal polygon, lying in parallel planes. A special case is a regular quadrangular prism. Its bases are 2 identical regular quadrangles, to which the sides are perpendicular, having the shape of parallelograms (or rectangles, if the prism is not inclined).
What does a prism look like?
A regular quadrangular prism is a hexagon, the bases of which are 2 squares, and the side faces are represented by rectangles. Another name for this geometric figure- straight parallelepiped.
A drawing showing a quadrangular prism is shown below.
You can also see in the picture essential elements, of which it consists geometric body . These include:
Sometimes in geometry problems you can come across the concept of a section. The definition will sound like this: a section is all the points of a volumetric body belonging to a cutting plane. The section can be perpendicular (intersects the edges of the figure at an angle of 90 degrees). For a rectangular prism, a diagonal section is also considered (the maximum number of sections that can be constructed is 2), passing through 2 edges and the diagonals of the base.
If the section is drawn in such a way that the cutting plane is not parallel to either the bases or the side faces, the result is a truncated prism.
To find the given prismatic elements, various relations and formulas are used. Some of them are known from the planimetry course (for example, to find the area of the base of a prism, it is enough to remember the formula for the area of a square).
Surface area and volume
To determine the volume of a prism using the formula, you need to know the area of its base and height:
V = Sbas h
Since the base of a regular tetrahedral prism is a square with side a, You can write the formula in more detailed form:
V = a²·h
If we are talking about a cube - a regular prism with equal length, width and height, the volume is calculated as follows:
To understand how to find the lateral surface area of a prism, you need to imagine its development.
From the drawing it can be seen that the side surface is made up of 4 equal rectangles. Its area is calculated as the product of the perimeter of the base and the height of the figure:
Sside = Posn h
Taking into account that the perimeter of the square is equal to P = 4a, the formula takes the form:
Sside = 4a h
For cube:
Sside = 4a²
To calculate the total surface area of the prism, you need to add 2 base areas to the lateral area:
Sfull = Sside + 2Smain
In relation to a quadrangular regular prism, the formula looks like:
Stotal = 4a h + 2a²
For the surface area of a cube:
Sfull = 6a²
Knowing the volume or surface area, you can calculate the individual elements of a geometric body.
Finding prism elements
Often there are problems in which the volume is given or the value of the lateral surface area is known, where it is necessary to determine the length of the side of the base or the height. In such cases, the formulas can be derived:
- base side length: a = Sside / 4h = √(V / h);
- height or side rib length: h = Sside / 4a = V / a²;
- base area: Sbas = V / h;
- side face area: Side gr = Sside / 4.
To determine how much area the diagonal section has, you need to know the length of the diagonal and the height of the figure. For a square d = a√2. From this it follows:
Sdiag = ah√2
To calculate the diagonal of a prism, use the formula:
dprize = √(2a² + h²)
To understand how to apply the given relationships, you can practice and solve several simple tasks.
Examples of problems with solutions
Here are some tasks found on state final exams in mathematics.
Task 1.
Sand is poured into a box shaped like a regular quadrangular prism. The height of its level is 10 cm. What will the sand level be if you move it into a container of the same shape, but with a base twice as long?
It should be reasoned as follows. The amount of sand in the first and second containers did not change, i.e. its volume in them is the same. You can denote the length of the base by a. In this case, for the first box the volume of the substance will be:
V₁ = ha² = 10a²
For the second box, the length of the base is 2a, but the height of the sand level is unknown:
V₂ = h (2a)² = 4ha²
Because V₁ = V₂, we can equate the expressions:
10a² = 4ha²
After reducing both sides of the equation by a², we get:
As a result, the new sand level will be h = 10 / 4 = 2.5 cm.
Task 2.
ABCDA₁B₁C₁D₁ is a correct prism. It is known that BD = AB₁ = 6√2. Find the total surface area of the body.
To make it easier to understand which elements are known, you can draw a figure.
Since we are talking about a regular prism, we can conclude that at the base there is a square with a diagonal of 6√2. The diagonal of the side face has the same size, therefore, the side face also has the shape of a square equal to the base. It turns out that all three dimensions - length, width and height - are equal. We can conclude that ABCDA₁B₁C₁D₁ is a cube.
The length of any edge is determined through a known diagonal:
a = d / √2 = 6√2 / √2 = 6
The total surface area is found using the formula for a cube:
Sfull = 6a² = 6 6² = 216
Task 3.
The room is being renovated. It is known that its floor has the shape of a square with an area of 9 m². The height of the room is 2.5 m. What is the lowest cost of wallpapering a room if 1 m² costs 50 rubles?
Since the floor and ceiling are squares, i.e. regular quadrangles, and its walls are perpendicular to horizontal surfaces, we can conclude that it is a regular prism. It is necessary to determine the area of its lateral surface.
The length of the room is a = √9 = 3 m.
The area will be covered with wallpaper Sside = 4 3 2.5 = 30 m².
The lowest cost of wallpaper for this room will be 50·30 = 1500 rubles
Thus, to solve problems involving a rectangular prism, it is enough to be able to calculate the area and perimeter of a square and rectangle, as well as to know the formulas for finding the volume and surface area.
How to find the area of a cube
A prism is a geometric three-dimensional figure, the characteristics and properties of which are studied in high schools. As a rule, when studying it, quantities such as volume and surface area are considered. In this article we will discuss a slightly different question: we will present a method for determining the length of the diagonals of a prism using the example of a quadrangular figure.
What shape is called a prism?
In geometry, the following definition of a prism is given: it is a three-dimensional figure bounded by two polygonal identical sides that are parallel to each other and a certain number of parallelograms. The figure below shows an example of a prism corresponding to this definition.
We see that the two red pentagons are equal to each other and are in two parallel planes. Five pink parallelograms connect these pentagons into a solid object - a prism. The two pentagons are called the bases of the figure, and its parallelograms are the side faces.
Prisms can be straight or oblique, also called rectangular or oblique. The difference between them lies in the angles between the base and the side edges. For a rectangular prism, all these angles are equal to 90 o.
Based on the number of sides or vertices of the polygon at the base, they speak of triangular, pentagonal, quadrangular prisms, and so on. Moreover, if this polygon is regular, and the prism itself is straight, then such a figure is called regular.
The prism shown in the previous figure is a pentagonal inclined one. Below is a pentagonal right prism, which is regular.
It is convenient to perform all calculations, including the method for determining the diagonals of a prism, specifically for the correct figures.
What elements characterize a prism?
The elements of a figure are the components that form it. Specifically for a prism, three main types of elements can be distinguished:
- tops;
- edges or sides;
- ribs
Faces are considered to be the bases and lateral planes, representing parallelograms in the general case. In a prism, each side is always one of two types: either it is a polygon or a parallelogram.
The edges of a prism are those segments that limit each side of the figure. Like faces, edges also come in two types: those belonging to the base and side surface or those belonging only to the side surface. There are always twice as many of the former as of the latter, regardless of the type of prism.
The vertices are the intersection points of three edges of the prism, two of which lie in the plane of the base, and the third belongs to the two lateral faces. All the vertices of the prism are in the planes of the bases of the figure.
The numbers of the described elements are connected into a single equality, which has the following form:
P = B + C - 2.
Here P is the number of edges, B - vertices, C - sides. This equality is called Euler's theorem for the polyhedron.
The figure shows a triangular regular prism. Everyone can count that it has 6 vertices, 5 sides and 9 edges. These figures are consistent with Euler's theorem.
Prism diagonals
After such properties as volume and surface area, in geometry problems we often encounter information about the length of a particular diagonal of the figure in question, which is either given or needs to be found using other known parameters. Let's consider what diagonals a prism has.
All diagonals can be divided into two types:
- Lying in the plane of the faces. They connect non-adjacent vertices of either a polygon at the base of a prism or a parallelogram on the lateral surface. The value of the lengths of such diagonals is determined based on knowledge of the lengths of the corresponding edges and the angles between them. To determine the diagonals of parallelograms, the properties of triangles are always used.
- Prisms lying inside the volume. These diagonals connect the dissimilar vertices of two bases. These diagonals are completely inside the figure. Their lengths are somewhat more difficult to calculate than for the previous type. The calculation method involves taking into account the lengths of the ribs and the base, and parallelograms. For straight and regular prisms the calculation is relatively simple as it is carried out using the Pythagorean theorem and the properties of trigonometric functions.
Diagonals of the sides of a quadrangular right prism
The figure above shows four identical straight prisms and gives the parameters of their edges. On the Diagonal A, Diagonal B and Diagonal C prisms, the dashed red line shows the diagonals of three different faces. Since the prism is a straight line with a height of 5 cm, and its base is represented by a rectangle with sides of 3 cm and 2 cm, it is not difficult to find the marked diagonals. To do this, you need to use the Pythagorean theorem.
The length of the diagonal of the base of the prism (Diagonal A) is equal to:
D A = √(3 2 +2 2) = √13 ≈ 3.606 cm.
For the side face of the prism, the diagonal is equal (see Diagonal B):
D B = √(3 2 +5 2) = √34 ≈ 5.831 cm.
Finally, the length of another side diagonal is (see Diagonal C):
D C = √(2 2 +5 2) = √29 ≈ 5.385 cm.
Inner diagonal length
Now let's calculate the length of the diagonal of the quadrangular prism, which is shown in the previous figure (Diagonal D). This is not so difficult to do if you notice that it is the hypotenuse of a triangle in which the legs will be the height of the prism (5 cm) and the diagonal D A shown in the figure at the top left (Diagonal A). Then we get:
D D = √(D A 2 +5 2) = √(2 2 +3 2 +5 2) = √38 ≈ 6.164 cm.
Regular quadrangular prism
The diagonal of a regular prism, the base of which is a square, is calculated in the same way as in the example above. The corresponding formula is:
D = √(2*a 2 +c 2).
Where a and c are the lengths of the side of the base and the side edge, respectively.
Note that in the calculations we used only the Pythagorean theorem. To determine the lengths of the diagonals of regular prisms with a large number vertices (pentagonal, hexagonal, etc.) it is already necessary to apply trigonometric functions.
Stereometry is an important part general course geometry, which examines the characteristics of spatial figures. One such figure is a quadrangular prism. In this article we will discuss in more detail the question of how to calculate the volume of a quadrangular prism.
What is a quadrangular prism?
Obviously, before giving the formula for the volume of a quadrangular prism, it is necessary to give a clear definition of this geometric figure. By such a prism we mean a three-dimensional polyhedron, which is limited by two arbitrary identical quadrangles lying in parallel planes and four parallelograms.
The marked quadrilaterals parallel to each other are called the bases of the figure, and the four parallelograms are the sides. It should be clarified here that parallelograms are also quadrilaterals, but the bases are not always parallelograms. An example of an irregular quadrilateral, which may well be the base of a prism, is shown in the figure below.
Any quadrangular prism consists of 6 sides, 8 vertices and 12 edges. There are quadrangular prisms different types. For example, a figure can be oblique or straight, irregular and regular. Later in the article we will show how you can calculate the volume of a quadrangular prism, taking into account its type.
Inclined prism with incorrect base
This is the most asymmetrical type of quadrangular prism, so calculating its volume will be relatively difficult. The following expression allows you to determine the volume of a figure:
The symbol So here denotes the area of the base. If this base is a rhombus, parallelogram or rectangle, then calculating the value of So is easy. So, for a rhombus and a parallelogram the formula is valid:
where a is the side of the base, ha is the length of the height lowered to this side from the top of the base.
If the base is an irregular polygon (see above), then its area should be divided into simpler shapes (for example, triangles), calculate their areas and find their sum.
In the formula for volume, the symbol h denotes the height of the prism. It represents the length of the perpendicular segment between two bases. Since the prism is inclined, the height h should be calculated using the length of the side edge b and the dihedral angles between the side faces and the base.
The correct figure and its volume
If the base of a quadrangular prism is a square, and the figure itself is straight, then it is called regular. It should be clarified that a straight prism is called when all its sides are rectangles and each of them is perpendicular to the bases. The correct figure is shown below.
The volume of a regular quadrangular prism can be calculated using the same formula as the volume of an irregular figure. Since the base is a square, its area is calculated simply:
The height of the prism h is equal to the length of the side edge b (side of the rectangle). Then the volume of a regular quadrangular prism can be calculated using the following formula:
A regular prism with a square base is called a rectangular parallelepiped. If sides a and b are equal, this parallelepiped becomes a cube. The volume of the latter is calculated as follows:
The written formulas for volume V indicate that the higher the symmetry of the figure, the fewer linear parameters are required to calculate this value. So, in the case of a regular prism, the required number of parameters is two, and in the case of a cube - one.
The problem with the correct figure
Having considered the issue of finding the volume of a quadrangular prism from a theoretical point of view, we will apply the acquired knowledge in practice.
It is known that a regular parallelepiped has a base diagonal length of 12 cm. The diagonal length of its side is 20 cm. It is necessary to calculate the volume of the parallelepiped.
Let us denote the diagonal of the base by the symbol da, and the diagonal of the side face by the symbol db. For the diagonal da the following expressions are valid:
As for the value db, it is the diagonal of a rectangle with sides a and b. For it we can write the following equalities:
db2 = a2 + b2 =>
b = √(db2 - a2)
Substituting the found expression for a into the last equality, we obtain:
b = √(db2 - da2/2)
Now you can substitute the resulting formulas into the expression for the volume of the regular figure:
V = a2*b = da2/2*√(db2 - da2/2)
Replacing da and db with the numbers from the problem statement, we arrive at the answer: V ≈ 1304 cm3.
Different prisms are different from each other. At the same time, they have a lot in common. To find the area of the base of the prism, you will need to understand what type it has.
General theory
A prism is any polyhedron whose sides have the shape of a parallelogram. Moreover, its base can be any polyhedron - from a triangle to an n-gon. Moreover, the bases of the prism are always equal to each other. What does not apply to the side faces is that they can vary significantly in size.
When solving problems, not only the area of the base of the prism is encountered. It may require knowledge of the lateral surface, that is, all the faces that are not bases. Full surface there will already be a union of all the faces that make up the prism.
Sometimes problems involve height. It is perpendicular to the bases. The diagonal of a polyhedron is a segment that connects in pairs any two vertices that do not belong to the same face.
It should be noted that the base area of a straight or inclined prism does not depend on the angle between them and the side faces. If they have the same figures on the top and bottom faces, then their areas will be equal.
Triangular prism
It has at its base a figure with three vertices, that is, a triangle. As you know, it can be different. If so, it is enough to remember that its area is determined by half the product of the legs.
The mathematical notation looks like this: S = ½ av.
To find out the area of the base in general view, the formulas will be useful: Heron and the one in which half of the side is taken to the height drawn to it.
The first formula should be written as follows: S = √(р (р-а) (р-в) (р-с)). This notation contains a semi-perimeter (p), that is, the sum of three sides divided by two.
Second: S = ½ n a * a.
If you want to find out the area of the base of a triangular prism, which is regular, then the triangle turns out to be equilateral. There is a formula for it: S = ¼ a 2 * √3.
Quadrangular prism
Its base is any of the known quadrangles. It can be a rectangle or square, parallelepiped or rhombus. In each case, in order to calculate the area of the base of the prism, you will need your own formula.
If the base is a rectangle, then its area is determined as follows: S = ab, where a, b are the sides of the rectangle.
When it comes to a quadrangular prism, the area of the base of a regular prism is calculated using the formula for a square. Because it is he who lies at the foundation. S = a 2.
In the case when the base is a parallelepiped, the following equality will be needed: S = a * n a. It happens that the side of a parallelepiped and one of the angles are given. Then, to calculate the height, you will need to use an additional formula: n a = b * sin A. Moreover, angle A is adjacent to side “b”, and height n is opposite to this angle.
If there is a rhombus at the base of the prism, then to determine its area you will need the same formula as for a parallelogram (since it is a special case of it). But you can also use this: S = ½ d 1 d 2. Here d 1 and d 2 are two diagonals of the rhombus.
Regular pentagonal prism
This case involves dividing the polygon into triangles, the areas of which are easier to find out. Although it happens that figures can have a different number of vertices.
Since the base of the prism is a regular pentagon, it can be divided into five equilateral triangles. Then the area of the base of the prism is equal to the area of one such triangle (the formula can be seen above), multiplied by five.
Regular hexagonal prism
According to the principle described for a pentagonal prism, it is possible to divide the hexagon of the base into 6 equilateral triangles. The formula for the base area of such a prism is similar to the previous one. Only it should be multiplied by six.
The formula will look like this: S = 3/2 a 2 * √3.
Tasks
No. 1. Given a regular straight line, its diagonal is 22 cm, the height of the polyhedron is 14 cm. Calculate the area of the base of the prism and the entire surface.
Solution. The base of the prism is a square, but its side is unknown. You can find its value from the diagonal of the square (x), which is related to the diagonal of the prism (d) and its height (h). x 2 = d 2 - n 2. On the other hand, this segment “x” is the hypotenuse in a triangle whose legs are equal to the side of the square. That is, x 2 = a 2 + a 2. Thus it turns out that a 2 = (d 2 - n 2)/2.
Substitute the number 22 instead of d, and replace “n” with its value - 14, it turns out that the side of the square is 12 cm. Now just find out the area of the base: 12 * 12 = 144 cm 2.
To find out the area of the entire surface, you need to add twice the base area and quadruple the side area. The latter can be easily found using the formula for a rectangle: multiply the height of the polyhedron and the side of the base. That is, 14 and 12, this number will be equal to 168 cm 2. Total area The surface of the prism turns out to be 960 cm 2.
Answer. The area of the base of the prism is 144 cm 2. The entire surface is 960 cm 2.
No. 2. Given At the base there is a triangle with a side of 6 cm. In this case, the diagonal of the side face is 10 cm. Calculate the areas: the base and the side surface.
Solution. Since the prism is regular, its base is equilateral triangle. Therefore, its area turns out to be equal to 6 squared, multiplied by ¼ and by the square root of 3. A simple calculation leads to the result: 9√3 cm 2. This is the area of one base of the prism.
All side faces are the same and are rectangles with sides of 6 and 10 cm. To calculate their areas, just multiply these numbers. Then multiply them by three, because the prism has exactly that many side faces. Then the area of the lateral surface of the wound turns out to be 180 cm 2.
Answer. Areas: base - 9√3 cm 2, lateral surface of the prism - 180 cm 2.
