Prism. Parallelepiped

prism is called a polyhedron whose two faces are equal n-gons (grounds) , lying in parallel planes, and the remaining n faces are parallelograms (side faces) . Side rib prism is the side of the lateral face that does not belong to the base.

A prism whose lateral edges are perpendicular to the planes of the bases is called straight prism (Fig. 1). If the side edges are not perpendicular to the planes of the bases, then the prism is called oblique . Correct A prism is a straight prism whose bases are regular polygons.

Height prism is called the distance between the planes of the bases. Diagonal A prism is a segment connecting two vertices that do not belong to the same face. diagonal section A section of a prism by a plane passing through two side edges that do not belong to the same face is called. Perpendicular section called the section of the prism by a plane perpendicular to the lateral edge of the prism.

Side surface area prism is the sum of the areas of all side faces. Full surface area the sum of the areas of all the faces of the prism is called (i.e., the sum of the areas of the side faces and the areas of the bases).

For an arbitrary prism, the formulas are true:

Where l is the length of the side rib;

H- height;

P

Q

S side

S full

S main is the area of ​​the bases;

V is the volume of the prism.

For a straight prism, the following formulas are true:

Where p- the perimeter of the base;

l is the length of the side rib;

H- height.

Parallelepiped A prism whose base is a parallelogram is called. A parallelepiped whose lateral edges are perpendicular to the bases is called direct (Fig. 2). If the side edges are not perpendicular to the bases, then the parallelepiped is called oblique . A right parallelepiped whose base is a rectangle is called rectangular. A rectangular parallelepiped in which all edges are equal is called cube.

The faces of a parallelepiped that do not have common vertices are called opposite . The lengths of edges emanating from one vertex are called measurements parallelepiped. Since the box is a prism, its main elements are defined in the same way as they are defined for prisms.

Theorems.

1. The diagonals of the parallelepiped intersect at one point and bisect it.

2. In a rectangular parallelepiped, the square of the length of the diagonal is equal to the sum of the squares of its three dimensions:

3. All four diagonals of a rectangular parallelepiped are equal to each other.

For an arbitrary parallelepiped, the following formulas are true:

Where l is the length of the side rib;

H- height;

P is the perimeter of the perpendicular section;

Q– Area of ​​perpendicular section;

S side is the lateral surface area;

S full is the total surface area;

S main is the area of ​​the bases;

V is the volume of the prism.

For a right parallelepiped, the following formulas are true:

Where p- the perimeter of the base;

l is the length of the side rib;

H is the height of the right parallelepiped.

For a rectangular parallelepiped, the following formulas are true:

(3)

Where p- the perimeter of the base;

H- height;

d- diagonal;

a,b,c– measurements of a parallelepiped.

The correct formulas for a cube are:

Where a is the length of the rib;

d is the diagonal of the cube.

Example 1 The diagonal of a rectangular cuboid is 33 dm, and its measurements are related as 2:6:9. Find the measurements of the cuboid.

Solution. To find the dimensions of the parallelepiped, we use formula (3), i.e. the fact that the square of the hypotenuse of a cuboid is equal to the sum of the squares of its dimensions. Denote by k coefficient of proportionality. Then the dimensions of the parallelepiped will be equal to 2 k, 6k and 9 k. We write formula (3) for the problem data:

Solving this equation for k, we get:

Hence, the dimensions of the parallelepiped are 6 dm, 18 dm and 27 dm.

Answer: 6 dm, 18 dm, 27 dm.

Example 2 Find the volume of an inclined triangular prism whose base is an equilateral triangle with a side of 8 cm, if the lateral edge is equal to the side of the base and is inclined at an angle of 60º to the base.

Solution . Let's make a drawing (Fig. 3).

In order to find the volume of an inclined prism, you need to know the area of ​​\u200b\u200bits base and height. The area of ​​the base of this prism is the area of ​​an equilateral triangle with a side of 8 cm. Let's calculate it:

The height of a prism is the distance between its bases. From the top A 1 of the upper base we lower the perpendicular to the plane of the lower base A 1 D. Its length will be the height of the prism. Consider D A 1 AD: since this is the angle of inclination of the side rib A 1 A to the base plane A 1 A= 8 cm. From this triangle we find A 1 D:

Now we calculate the volume using formula (1):

Answer: 192 cm3.

Example 3 The lateral edge of a regular hexagonal prism is 14 cm. The area of ​​\u200b\u200bthe largest diagonal section is 168 cm 2. Find the total surface area of ​​the prism.

Solution. Let's make a drawing (Fig. 4)

The largest diagonal section is a rectangle AA 1 DD 1 , since the diagonal AD regular hexagon ABCDEF is the largest. In order to calculate the lateral surface area of ​​a prism, it is necessary to know the side of the base and the length of the lateral rib.

Knowing the area of ​​the diagonal section (rectangle), we find the diagonal of the base.

Since , then

Since then AB= 6 cm.

Then the perimeter of the base is:

Find the area of ​​the lateral surface of the prism:

The area of ​​a regular hexagon with a side of 6 cm is:

Find the total surface area of ​​the prism:

Answer:

Example 4 The base of a right parallelepiped is a rhombus. The areas of diagonal sections are 300 cm 2 and 875 cm 2. Find the area of ​​the side surface of the parallelepiped.

Solution. Let's make a drawing (Fig. 5).

Denote the side of the rhombus by A, the diagonals of the rhombus d 1 and d 2 , the height of the box h. To find the lateral surface area of ​​a straight parallelepiped, it is necessary to multiply the perimeter of the base by the height: (formula (2)). Base perimeter p = AB + BC + CD + DA = 4AB = 4a, because ABCD- rhombus. H = AA 1 = h. That. Need to find A And h.

Consider diagonal sections. AA 1 SS 1 - a rectangle, one side of which is the diagonal of a rhombus AU = d 1 , second - side edge AA 1 = h, Then

Similarly for the section BB 1 DD 1 we get:

Using the property of a parallelogram such that the sum of the squares of the diagonals is equal to the sum of the squares of all its sides, we get the equality We get the following.

Definition.

This is a hexagon, the bases of which are two equal squares, and the side faces are equal rectangles.

Side rib is the common side of two adjacent side faces

Prism Height is a line segment perpendicular to the bases of the prism

Prism Diagonal- a segment connecting two vertices of the bases that do not belong to the same face

Diagonal plane- a plane that passes through the diagonal of the prism and its side edges

Diagonal section- the boundaries of the intersection of the prism and the diagonal plane. The diagonal section of a regular quadrangular prism is a rectangle

Perpendicular section (orthogonal section)- this is the intersection of a prism and a plane drawn perpendicular to its side edges

Elements of a regular quadrangular prism

The figure shows two regular quadrangular prisms, which are marked with the corresponding letters:

• Bases ABCD and A 1 B 1 C 1 D 1 are equal and parallel to each other
• Side faces AA 1 D 1 D, AA 1 B 1 B, BB 1 C 1 C and CC 1 D 1 D, each of which is a rectangle
• Lateral surface - the sum of the areas of all the side faces of the prism
• Total surface - the sum of the areas of all bases and side faces (the sum of the area of ​​the side surface and bases)
• Side ribs AA 1 , BB 1 , CC 1 and DD 1 .
• Diagonal B 1 D
• Base diagonal BD
• Diagonal section BB 1 D 1 D
• Perpendicular section A 2 B 2 C 2 D 2 .

Properties of a regular quadrangular prism

• The bases are two equal squares
• The bases are parallel to each other
• The sides are rectangles.
• Side faces are equal to each other
• Side faces are perpendicular to the bases
• Lateral ribs are parallel to each other and equal
• Perpendicular section perpendicular to all side ribs and parallel to the bases
• Perpendicular Section Angles - Right
• The diagonal section of a regular quadrangular prism is a rectangle
• Perpendicular (orthogonal section) parallel to the bases

Formulas for a regular quadrangular prism

Instructions for solving problems

When solving problems on the topic " regular quadrangular prism" implies that:

Correct prism- a prism at the base of which lies a regular polygon, and the side edges are perpendicular to the planes of the base. That is, a regular quadrangular prism contains at its base square. (see above the properties of a regular quadrangular prism) Note. This is part of the lesson with tasks in geometry (section solid geometry - prism). Here are the tasks that cause difficulties in solving. If you need to solve a problem in geometry, which is not here - write about it in the forum. To denote the action of extracting a square root in solving problems, the symbol is used√ .

Task.

In a regular quadrangular prism, the base area is 144 cm 2 and the height is 14 cm. Find the diagonal of the prism and the total surface area.

Solution.
A regular quadrilateral is a square.
Accordingly, the side of the base will be equal to

√144 = 12 cm.
Whence the diagonal of the base of a regular rectangular prism will be equal to
√(12 2 + 12 2 ) = √288 = 12√2

The diagonal of a regular prism forms a right triangle with the diagonal of the base and the height of the prism. Accordingly, according to the Pythagorean theorem, the diagonal of a given regular quadrangular prism will be equal to:
√((12√2) 2 + 14 2 ) = 22 cm

Answer: 22 cm

Task

Find the total surface area of ​​a regular quadrangular prism if its diagonal is 5 cm and the diagonal of the side face is 4 cm.

Solution.
Since the base of a regular quadrangular prism is a square, then the side of the base (denoted as a) is found by the Pythagorean theorem:

A 2 + a 2 = 5 2
2a 2 = 25
a = √12.5

The height of the side face (denoted as h) will then be equal to:

H 2 + 12.5 \u003d 4 2
h 2 + 12.5 = 16
h 2 \u003d 3.5
h = √3.5

The total surface area will be equal to the sum of the lateral surface area and twice the base area

S = 2a 2 + 4ah
S = 25 + 4√12.5 * √3.5
S = 25 + 4√43.75
S = 25 + 4√(175/4)
S = 25 + 4√(7*25/4)
S \u003d 25 + 10√7 ≈ 51.46 cm 2.

Answer: 25 + 10√7 ≈ 51.46 cm 2.

In spatial geometry, when solving problems with prisms, there is often a problem with calculating the area of ​​the sides or faces that form these three-dimensional figures. This article is devoted to the issue of determining the area of ​​the base of the prism and its lateral surface.

Figure prism

Before proceeding to the consideration of the formulas for the area of ​​​​the base and the surface of a prism of one kind or another, it is necessary to understand what kind of figure we are talking about.

A prism in geometry is a spatial figure consisting of two parallel polygons that are equal to each other, and several quadrangles or parallelograms. The number of the latter is always equal to the number of vertices of one polygon. For example, if the figure is formed by two parallel n-gons, then the number of parallelograms will be n.

The connecting n-gons of the parallelogram are called the sides of the prism, and their total area is the area of ​​the side surface of the figure. The n-gons themselves are called bases.

The figure above shows an example of a paper prism. The yellow rectangle is its upper base. On the second base of the same figure stands. The red and green rectangles are the side faces.

What are the prisms?

There are several types of prisms. All of them differ from each other in just two parameters:

• the type of n-gon forming the bases;
• angle between the n-gon and the side faces.

For example, if the bases are triangles, then the prism is called triangular, if quadrilaterals, as in the previous figure, then the figure is called a quadrangular prism, and so on. In addition, the n-gon can be convex or concave, then this property is also added to the name of the prism.

The angle between the side faces and the base can be either straight or acute or obtuse. In the first case, they talk about a rectangular prism, in the second - about an inclined or oblique.

Regular prisms are distinguished into a special type of figure. They have the highest symmetry among the other prisms. It will be correct only if it is rectangular and its base is a regular n-gon. The figure below shows a set of regular prisms, in which the number of sides of the n-gon varies from three to eight.

Prism surface

Under the surface of the considered figure of an arbitrary type is understood the totality of all points that belong to the faces of the prism. It is convenient to study the surface of a prism by considering its development. Below is an example of such a sweep for a triangular prism.

It can be seen that the entire surface is formed by two triangles and three rectangles.

In the case of a general type prism, its surface will consist of two n-gonal bases and n quadrilaterals.

Let us consider in more detail the issue of calculating the surface area of ​​prisms of different types.

Base area of ​​a prism

Perhaps the easiest task when working with prisms is the problem of finding the base area of ​​a regular figure. Since it is formed by an n-gon, in which all angles and side lengths are the same, it is always possible to divide it into identical triangles, for which angles and sides are known. The total area of ​​the triangles will be the area of ​​the n-gon.

Another way to determine the portion of the surface area of ​​a prism (base) is to use a well-known formula. It looks like this:

S n = n/4*a 2 *ctg(pi/n)

That is, the area S n of an n-gon is uniquely determined based on the knowledge of the length of its side a. Some difficulty in calculating the formula can be the calculation of the cotangent, especially when n>4 (for n≤4, the values ​​of the cotangent are tabular data). To determine this trigonometric function, it is recommended to use a calculator.

When setting a geometric problem, you should be careful, because you may need to find the area of ​​​​the bases of the prism. Then the value obtained by the formula should be multiplied by two.

Base area of ​​a triangular prism

Using the example of a triangular prism, consider how you can find the area of ​​\u200b\u200bthe base of this figure.

First, consider a simple case - a regular prism. The area of ​​\u200b\u200bthe base is calculated according to the formula given in the paragraph above, you need to substitute n \u003d 3 into it. We get:

S 3 = 3/4*a 2 *ctg(pi/3) = 3/4*a 2 *1/√3 = √3/4*a 2

It remains to substitute in the expression the specific values ​​of the length of the side a of an equilateral triangle to get the area of ​​\u200b\u200bone base.

Now suppose we have a prism whose base is an arbitrary triangle. Its two sides a and b and the angle between them α are known. This figure is shown below.

How to find the area of ​​the base of a triangular prism in this case? It must be remembered that the area of ​​any triangle is equal to half the product of the side and the height lowered to this side. The figure shows the height h to side b. The length h corresponds to the product of the sine of the angle alpha and the length of the side a. Then the area of ​​the entire triangle is:

S = 1/2*b*h = 1/2*b*a*sin(α)

This is the base area of ​​the depicted triangular prism.

Side surface

We figured out how to find the area of ​​​​the base of a prism. The lateral surface of this figure always consists of parallelograms. For straight prisms, parallelograms become rectangles, so it's easy to calculate their total area:

S = ∑ i=1 n (a i *b)

Here b is the length of the side edge, and i is the length of the side of the i-th rectangle, which coincides with the length of the side of the n-gon. In the case of a regular n-gonal prism, we get a simple expression:

If the prism is inclined, then to determine the area of ​​its lateral surface, a perpendicular cut should be made, its perimeter P sr calculated and multiplied by the length of the lateral rib.

The figure above shows how this cut should be made for an oblique pentagonal prism.

Different prisms are different from each other. At the same time, they have a lot in common. To find the area of ​​\u200b\u200bthe base of a prism, you need to figure out what kind it looks like.

General theory

A prism is any polyhedron whose sides have the form of a parallelogram. Moreover, any polyhedron can be at its base - 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 - they can vary significantly in size.

When solving problems, it is not only the area of ​​\u200b\u200bthe base of the prism that is encountered. It may be necessary to know the lateral surface, that is, all faces that are not bases. The full surface will already be the union of all the faces that make up the prism.

Sometimes heights appear in tasks. 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 area of ​​the base of a straight or inclined prism does not depend on the angle between them and the side faces. If they have the same figures in the upper and lower faces, then their areas will be equal.

triangular prism

It has at the base a figure with three vertices, that is, a triangle. It is known to be different. If then it is enough to recall that its area is determined by half the product of the legs.

Mathematical notation looks like this: S = ½ av.

To find out the area of ​​\u200b\u200bthe base in a general form, the formulas are 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 like this: S \u003d √ (p (p-a) (p-in) (p-s)). This entry 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 know the area of ​​\u200b\u200bthe base of a triangular prism, which is regular, then the triangle turns out to be equilateral. It has its own formula: S = ¼ a 2 * √3.

quadrangular prism

Its base is any of the known quadrilaterals. It can be a rectangle or a square, a parallelepiped or a rhombus. In each case, in order to calculate the area of ​​\u200b\u200bthe base of the prism, you will need your own formula.

If the base is a rectangle, then its area is determined as follows: S = av, where a, b are the sides of the rectangle.

When it comes to a quadrangular prism, the base area of ​​a regular prism is calculated using the formula for a square. Because it is he who lies at the base. S \u003d a 2.

In the case when the base is a parallelepiped, the following equality will be needed: S \u003d a * n a. It happens that a side of a parallelepiped and one of the angles are given. Then, to calculate the height, you will need to use an additional formula: na \u003d b * sin A. Moreover, the angle A is adjacent to the side "b", and the height is na opposite to this angle.

If a rhombus lies at the base of the prism, then the same formula will be needed to determine its area as for a parallelogram (since it is a special case of it). But you can also use this one: S = ½ d 1 d 2. Here d 1 and d 2 are two diagonals of the rhombus.

Regular pentagonal prism

This case involves splitting the polygon into triangles, the areas of which are easier to find out. Although it happens that the figures can be with 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 ​​\u200b\u200bthe 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 base hexagon into 6 equilateral triangles. The formula for the area of ​​​​the base of such a prism is similar to the previous one. Only in it should be multiplied by six.

The formula will look like this: S = 3/2 and 2 * √3.

Tasks

No. 1. A regular straight line is given. Its diagonal is 22 cm, the height of the polyhedron is 14 cm. Calculate the area of ​​\u200b\u200bthe base of the prism and the entire surface.

Solution. The base of a prism is a square, but its side is not known. 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 \u003d 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 \u003d a 2 + a 2. Thus, it turns out that a 2 \u003d (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 it’s easy to find out the base area: 12 * 12 \u003d 144 cm 2.

To find out the area of ​​\u200b\u200bthe entire surface, you need to add twice the value of the base area and quadruple the side. The latter is easy to find by 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. The total surface area of ​​the prism is found to be 960 cm 2 .

Answer. The base area of ​​the prism is 144 cm2. The entire surface - 960 cm 2 .

No. 2. Dana At the base lies 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 an equilateral triangle. Therefore, its area turns out to be equal to 6 squared times ¼ and 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, it is enough to multiply these numbers. Then multiply them by three, because the prism has exactly so many side faces. Then the area of ​​the side surface is wound 180 cm 2 .

Answer. Areas: base - 9√3 cm 2, side surface of the prism - 180 cm 2.

In the school curriculum for the course of solid geometry, the study of three-dimensional figures usually begins with a simple geometric body - a prism polyhedron. The role of its bases is performed by 2 equal polygons lying in parallel planes. A special case is a regular quadrangular prism. Its bases are 2 identical regular quadrilaterals, 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, at the bases of which there are 2 squares, and the side faces are represented by rectangles. Another name for this geometric figure is a straight parallelepiped.

The figure, which depicts a quadrangular prism, is shown below.

You can also see in the picture the most important elements that make up a geometric body. They are commonly referred to as:

Sometimes in problems in geometry you can find the concept of a section. The definition will sound like this: a section is all points of a volumetric body that belong to the cutting plane. The section is perpendicular (crosses 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 built 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.

Various ratios and formulas are used to find the reduced prismatic elements. Some of them are known from the course of planimetry (for example, to find the area of ​​the base of a prism, it is enough to recall 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 ​​\u200b\u200bits base and height:

V = Sprim h

Since the base of a regular tetrahedral prism is a square with side a, You can write the formula in a 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 sweep.

It can be seen from the drawing 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 = Pos h

Since the perimeter of a square is P = 4a, the formula takes the form:

Sside = 4a h

For cube:

Sside = 4a²

To calculate the total surface area of ​​a prism, add 2 base areas to the side area:

Sfull = Sside + 2Sbase

As applied to a quadrangular regular prism, the formula has the form:

Sfull = 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, formulas can be derived:

• base side length: a = Sside / 4h = √(V / h);
• height or side rib length: h = Sside / 4a = V / a²;
• base area: Sprim = V / h;
• side face area: Side gr = Sside / 4.

To determine how much area a diagonal section has, you need to know the length of the diagonal and the height of the figure. For a square d = a√2. Therefore:

Sdiag = ah√2

To calculate the diagonal of the prism, the formula is used:

dprize = √(2a² + h²)

To understand how to apply the above ratios, you can practice and solve a few simple tasks.

Examples of problems with solutions

Here are some of the tasks that appear in the state final exams in mathematics.

Exercise 1.

Sand is poured into a box shaped like a regular quadrangular prism. The height of its level is 10 cm. What will the level of sand be if you move it into a container of the same shape, but with a base length 2 times longer?

It should be argued 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 define the length of the base as 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 the V₁ = V₂, the expressions can be equated:

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 regular 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 the base is a square with a diagonal of 6√2. The diagonal of the side face has the same value, 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 the known diagonal:

a = d / √2 = 6√2 / √2 = 6

The total surface area is found by the formula for the 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, that is, 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 square 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 for a rectangular prism, it is enough to be able to calculate the area and perimeter of a square and a rectangle, as well as to know the formulas for finding the volume and surface area.

How to find the area of ​​a cube