How to calculate the total surface area of ​​a cube. How to find the area of ​​a cube

The cube has many interesting mathematical properties and has been known to people since ancient times. Representatives of some ancient Greek schools believed that elementary particles(atoms), of which our world consists, have the shape of a cube, and mystics and esotericists even deified this figure. And today, representatives of parascience attribute amazing energy properties to the cube.

The cube is ideal figure, one of the five Platonic solids. Platonic solid is

a regular polyhedral figure that satisfies three conditions:

1. All its edges and faces are equal.

2. The angles between the faces are equal (for a cube, the angles between the faces are equal and make up 90 degrees).

3. All vertices of the figure touch the surface of the sphere described around it.

The exact number of these figures called ancient Greek mathematician Theaetetus of Athens, and Plato's student Euclid, in the 13th book of the Beginnings, gave them a detailed mathematical description.

The ancient Greeks, who were inclined to describe the structure of our world with the help of quantitative quantities, gave the Platonic solids a deep sacred meaning. They believed that each of the figures symbolizes the universal principles: the tetrahedron - fire, the cube - earth, the octahedron - air, the icosahedron - water, the dodecahedron - ether. The sphere described around them symbolized perfection, the divine principle.

So, a cube, also called a hexahedron (from the Greek "hex" - 6), is a three-dimensional regular one. It is also called a rectangular parallelepiped.

A cube has six faces, twelve edges and eight vertices. Other tetrahedron (tetrahedron with triangle-shaped faces), octahedron (octahedron) and icosahedron (twenty-sided) can be inscribed in this figure.

A segment connecting two vertices symmetrical about the center is called. Knowing the length of the edge of the cube a, we can find the length of the diagonal v: v = a 3.

As mentioned above, a sphere can be inscribed in a cube, while the radius of the inscribed sphere (denoted by r) will be equal to half the length of the edge: r = (1/2) a.

If the sphere is described around a cube, then the radius of the circumscribed sphere (we denote it by R) will be equal to: R= (3/2)a.

A fairly common question in school problems: how to calculate the area

cube surface? It is very simple, it is enough to visualize a cube. The surface of a cube consists of six square-shaped faces. Therefore, in order to find the surface area of ​​a cube, you first need to find the area of ​​one of the faces and multiply by their number: S p \u003d 6a 2.

Similarly to how we found the surface area of ​​a cube, we calculate the area of ​​its side faces: S b = 4a 2.

From this formula, it is clear that the two opposite faces of the cube are the bases, and the remaining four are the side surfaces.

You can find the cube in another way. Given the fact that a cube is a rectangular parallelepiped, we can use the concept of three spatial dimensions. This means that the cube, being a three-dimensional figure, has 3 parameters: length (a), width (b) and height (c).

Using these parameters, we calculate the total surface area of ​​the cube: S p = 2(ab+ac+bc).

The volume of a cube is the product of three components - height, length and width:
V= abc or three adjacent edges: V=a 3.

This is the total area of ​​all the surfaces of the figure. The surface area of ​​a cube is equal to the sum of the areas of all its six faces. Surface area is a numerical characteristic of a surface. To calculate the surface area of ​​a cube, you need to know a certain formula and the length of one of the sides of the cube. In order for you to quickly calculate the surface area of ​​a cube, you need to remember the formula and the procedure itself. Below we will analyze in detail the order of calculation total surface area of ​​the cube and give specific examples.

It is carried out according to the formula SA \u003d 6a 2. The cube (regular hexahedron) is one of the 5 types of regular polyhedra, which is a regular rectangular parallelepiped, the cube has 6 faces, each of these faces is a square.

For calculating the surface area of ​​a cube You need to write down the formula SA = 6a 2 . Now let's see why given formula has this look. As we said earlier, a cube has six equal square faces. Based on the fact that the sides of the square are equal, the area of ​​the square is - a 2, where a is the side of the cube. Since a cube has 6 equal square faces, to determine its surface area, you need to multiply the area of ​​one face (square) by six. As a result, we obtain a formula for calculating the surface area (SA) of a cube: SA \u003d 6a 2, where a is the edge of the cube (side of the square).

What is the surface area of ​​a cube.

measured in square units, for example, in mm 2, cm 2, m 2 and so on. For further calculations, you will need to measure the edge of the cube. As we know, the edges of a cube are equal, so it will be enough for you to measure only one (any) edge of the cube. You can perform such a measurement using a ruler (or tape measure). Pay attention to the units of measure on the ruler or tape measure and write down the value, denoting it as a.

Example: a = 2 cm.

Square the resulting value. So you are squaring the edge length of the cube. To square a number, multiply it by itself. Our formula will look like this: SA \u003d 6 * a 2

You have calculated the area of ​​one of the faces of a cube.

Example: a = 2 cm

a 2 \u003d 2 x 2 \u003d 4 cm 2

Multiply the resulting value by six. Remember that a cube has 6 equal sides. Having determined the area of ​​one of the faces, multiply the resulting value by 6 so that all the faces of the cube are included in the calculation.

Here we come to the final action on calculating the surface area of ​​a cube.

Example: a 2 \u003d 4 cm 2

SA \u003d 6 x a 2 \u003d 6 x 4 \u003d 24 cm 2

Sharpen on the cube itself. It shows that any of the faces of the cube is a square. Thus, the problem of finding the area of ​​the face of a cube is reduced to the problem of finding the area of ​​any of the squares (faces of the cube). Any of the faces of the cube is possible, since the lengths of all its edges are between each other.

Example: The length of the edge of a cube is 11 cm, you need to find its area.

Solution: knowing the length of the face, you can find its area:

S = 11² = 121 cm²

Answer: the area of ​​the face of a cube with an edge of 11 cm is 121 cm²

note

Any cube has 8 vertices, 12 edges, 6 faces and 3 faces at the top.
The cube is such a figure that is incredibly common in everyday life. Enough to remember game dice, dice, cubes in various children's and teenage designers.
Many architectural elements are cubic in shape.
Cubic meters are used to measure volumes various substances in various areas of society.
talking scientific language, cubic meter is a measure of the volume of a substance that can fit in a cube with an edge length of 1 m
Thus, you can enter other units of volume: cubic millimeters, centimeters, decimeters, etc.
In addition to various cubic units of volume, in oil and gas industry it is possible to use another unit - barrel (1m³ = 6.29 barrels)

Helpful advice

If the length of its edge is known for a cube, then, in addition to the face area, other parameters of this cube can be found, for example:
Cube surface area: S = 6*a²;
Volume: V = 6*a³;
Radius of the inscribed sphere: r = a/2;
Radius of a sphere circumscribed around a cube: R = ((√3)*a))/2;
Diagonal of a cube (a segment connecting two opposite vertices of a cube that passes through its center): d = a*√3

Sources:

• area of ​​a cube if edges are 11 cm

A cube is a regular polyhedron, each face of which is a square. The area of ​​a cube is the area of ​​its surface, which consists of the sum of the areas of its faces, that is, the sum of the areas of the squares that form the cube.

The cube is one of the simplest three-dimensional shapes. Everyone is familiar with ice cubes, square boxes or salt crystals - they are all such figures. The surface area of ​​a cube is total area all sides on its surface. All six of its faces are commensurate, therefore, knowing the length of one of them, we can calculate side area and the surface area of ​​any figure.

How to find the area of ​​a cube - what is the figure?

A cube is a three-dimensional figure that has the same dimensions. Its length, width, and height are identical, and each edge meets the other edges at the same angle. Finding the surface area of ​​a cube is quick and easy because it is made up of congruent or commensurate squares. So, once you find the size of one of the squares, you will know the area of ​​the whole figure.

How to find the area of ​​a cube - faces of a figure

It can be seen from the illustration that the cube has a front and back face, two side faces and an upper one from the bottom side. The area of ​​any cube will be six congruent squares. In fact, if you expand it, you can clearly see the six squares that make up the overall surface of the figure.

How to find the area of ​​a cube

The area of ​​a cube is made up of the area of ​​six faces. Since they are all equal, it is enough to know the area of ​​​​one of them and multiply the value by 6. The area of ​​\u200b\u200bthe figure is also found using a simple formula: S \u003d 6 x a², where "a" is one of the sides of the cube.

How to find the area of ​​a cube - set the area of ​​a side

• Let's assume that the height of the cube is 2 cm. Since its surface is made up of squares, all its edges will have the same length. Therefore, based on the dimensions of the height, its length and width will be 2 cm.
• To find the area of ​​one of the squares, remember the basic knowledge of geometry, where S = a², where a is the length of one of the sides. In our case, a = 2 cm, so S = (2 cm)² = 2 cm x 2 cm = 4 cm².
• The area of ​​one of the surface squares is 4 cm². Be sure to include your value in square units.

How to find the area of ​​a cube - example

Since the entire surface of the figure consists of six proportional squares, you need to multiply the area of ​​\u200b\u200bone side by 6, following the formula S \u003d 6 x a². In our case, S = 6 x 4 cm² = 24 cm². The area of ​​a three-dimensional figure is 24 cm².

Find the area of ​​a cube if the side is in fractions

If you find it difficult to work with a fraction, convert it to a decimal.
For example, the height of a cube is 2 ½ cm.

• S = 6 x (2½ cm)²
• S = 6 x (2.5cm)²
• S = 6 x 6.25 cm²
• S = 37.5 cm²
• The surface area of ​​the cube is 37.5 cm².

Knowing the area of ​​a cube, find its side

If the surface area of ​​a cube is known, the length of its sides can be determined.

• The area of ​​a cube is 86.64 cm². You need to determine the length of the edge.
• Decision. Since the surface area is known, it is necessary to calculate in reverse order, dividing the value by 6, and then taking the square root.
• Having made the necessary calculations, we obtain a length of 3.8 cm.

How to find the area of ​​a cube - online area measurement

Using the calculator on the OnlineMSchool site, you can quickly calculate the area of ​​a cube. Enough to enter desired value parties and the service will issue a detailed step by step solution tasks.

So, to know the area of ​​a cube, calculate the area of ​​one of the sides, then multiply the result by 6 since the figure has 6 equal sides. You can use the formula S \u003d 6a² when calculating. If the surface area is given, it is possible to determine the length of the side part by doing the reverse steps.

Geometry is one of the main mathematical sciences, basic course which is taught even in school. In fact, the benefits of knowing various figures and laws will come in handy in everyone's life. Very often there are geometric problems on finding the area. If with flat figures students do not have any special problems, then voluminous may cause certain difficulties. Calculate cube surface area is not as simple as it seems at first glance. But with due attention, even the most difficult task is solved.

Necessary:

Knowledge of basic formulas;
- conditions of the problem.

Instruction:

• First of all, you need to decide which cube area formula is applicable in a particular case. For this you need to look at predefined figure parameters . What data is known: fin length, volume, diagonal, face area. Depending on this, the formula is selected.
• If, according to the conditions of the problem, it is known cube edge length, then it is enough to apply the simplest formula to find the area. Almost everyone knows that the area of ​​a square is found by multiplying the lengths of its two sides. faces of a cube- squares, therefore, its surface area is equal to the sum of the areas of these squares. A cube has six faces, so the formula for the area of ​​a cube would look like this: S=6*х 2 . Where X - cube edge length.
• Let's assume that cube edge not set, but known. Since the volume of a given figure is calculated by raising to the third power the length of its rib, then the latter can be obtained quite easily. To do this, from the number denoting the volume, it is necessary to extract the root of the third degree. For example, for the number 27 third root is the number 3 . Well, what to do next, we have already sorted it out. Thus, the formula for the area of ​​a cube with a known volume also exists, where instead of X is the third root of the volume.
• Sometimes known only diagonal length . If you remember the Pythagorean theorem, then we can easily calculate the length of the edge. Enough is here basic knowledge. The result obtained is substituted into the already known formula for the surface area of ​​a cube: S=6*х 2 .
• Summing up, it is worth noting that for correct calculations, you need to know the length of the edge. The conditions in the tasks are very different, so you should learn how to perform several actions at once. If other characteristics are known geometric figure, then with the help of additional formulas and theorems it is possible to calculate the edge of the cube. And already on the basis of the result, calculate the result.

A cube is a regular polyhedron, in which all faces are formed by regular quadrangles - squares. In order to find the area of ​​the face of any cube, heavy calculations are not required.

Instruction

To begin with, it is worth focusing on the very definition of a cube. It shows that any of the faces of the cube is a square. Thus, the problem of finding the area of ​​the face of a cube is reduced to the problem of finding the area of ​​any of the squares (faces of the cube). You can take exactly any of the faces of the cube, since the lengths of all its edges are equal to each other.

In order to find the area of ​​the face of a cube, you need to multiply a pair of any of its sides together, because they are all equal to each other. This can be expressed in a formula like this:

S = a?, where a is the side of the square (the edge of the cube).

Example: The length of the edge of a cube is 11 cm, you need to find its area.

Solution: knowing the length of the face, you can find its area:

S=11? = 121 cm?

Answer: the area of ​​the face of a cube with an edge of 11 cm is 121 cm?

note

Any cube has 8 vertices, 12 edges, 6 faces and 3 faces at the top.
The cube is such a figure that is incredibly common in everyday life. Suffice it to recall game cubes, dice, cubes in various children's and teenage designers.
Many architectural elements are cubic in shape.
Cubic meters are used to measure the volumes of various substances in various spheres of society.
In scientific terms, a cubic meter is a measure of the volume of a substance that can fit in a cube with an edge length of 1 m.
Thus, you can enter other units of volume: cubic millimeters, centimeters, decimeters, etc.
In addition to various cubic units of volume, in the oil and gas industry it is possible to use a different unit - barrel (1m? = 6.29 barrels)

Helpful advice

If the length of its edge is known for a cube, then, in addition to the face area, other parameters of this cube can be found, for example:
Cube surface area: S = 6*a?;
Volume: V = 6*a?;
Radius of the inscribed sphere: r = a/2;
Radius of a sphere circumscribed around a cube: R = ((?3)*a))/2;
Diagonal of a cube (a segment connecting two opposite vertices of a cube that passes through its center): d = a*?3