Find the edge of a cube if the area is known. Volumes of figures. Volume of a cube

Knowing some parameters of a cube, you can easily find its edge. To do this, it is enough just to have information about its volume, the area of ​​a face or the length of the diagonal of a face or cube.

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Instructions

There are mainly four types of problems in which you need to find the edge of a cube. This is the determination of the length of a cube edge by the area of ​​the cube face, by the volume of the cube, by the diagonal of the cube face and by the diagonal of the cube. Let's consider all four variants of such problems. (The remaining tasks, as a rule, are variations of the above or trigonometry tasks that are very indirectly related to the issue at hand)

If the area of ​​a cube face is known, then finding the edge of the cube is very simple. Since the face of a cube is a square with a side equal to the edge of the cube, its area is equal to the square of the edge of the cube. Therefore, the length of the edge of a cube is equal to the square root of the area of ​​its face, that is:

a is the length of the cube edge,

S is the area of ​​the cube face.

Finding the face of a cube based on its volume is even easier. Considering that the volume of a cube is equal to the cube (third power) of the length of the edge of the cube, we find that the length of the edge of the cube is equal to the cube root (third power) of its volume, i.e.:

a=?V ( cube root), Where

a is the length of the cube edge,

V is the volume of the cube.

It is a little more difficult to find the length of the edge of a cube using the known lengths of the diagonals. Let's denote by:

a is the length of the edge of the cube-

b - length of the diagonal of the cube face -

c is the length of the diagonal of the cube.

As can be seen from the figure, the diagonal of the face and the edges of the cube form a right-angled equilateral triangle. Therefore, according to the Pythagorean theorem:

(^ is the symbol for exponentiation).

From here we find:

(to find the edge of the cube you need to extract Square root from half the square of the diagonal of the face).

To find the edge of the cube along its diagonal, we will again use the figure. The diagonal of the cube (c), the diagonal of the face (b) and the edge of the cube (a) form right triangle. So, according to the Pythagorean theorem:

Let's use the above-established relationship between a and b and substitute it into the formula

b^2=a^2+a^2. We get:

a^2+a^2+a^2=c^2, from where we find:

3*a^2=c^2, therefore.

The problems presented below are simple, most of them can be solved in 1 step. In this article we will consider cuboid(all faces are rectangles). What do you need to know and understand? First, look at the formulas for the volume and surface area of ​​a cube and a rectangular parallelepiped, as well as the diagonal formula, you can.Let us briefly list the formulas:

Rectangular parallelepiped

Let the edges be equal A,b, With.

Surface area:

Volume:

Diagonal:

Cube

Let the edge of the cube be equal A.

Surface area:

Volume:

Diagonal:

*It is clear that the formulas of a cube are a consequence of the corresponding formulas of a rectangular parallelepiped. A cube is a parallelepiped in which all edges are equal and the faces are squares.

Let's consider the tasks:

The two edges of a cuboid coming from the same vertex are 5 and 8. The surface area of ​​this cuboid is 210. Find the third edge coming from the same vertex.

Let us denote the known edges as A And b, and the unknown for c.

Then the formula for the surface area of ​​a parallelepiped is expressed as:

All that remains is to substitute the data and solve the equation:

Answer: 5

The surface area of ​​a cube is 200. Find its diagonal.

Let's construct the diagonal of the cube:

The surface area of ​​a cube is expressed in terms of its edge A How S = 6A 2, which means we can find the edge A:

The diagonal of the face of a cube according to the Pythagorean theorem is equal to:

The diagonal of a cube according to the Pythagorean theorem is equal to:

Then

*You could immediately use the cube diagonal formula:

Answer: 10

The volume of the cube is 343. Find its surface area.

The surface area of ​​a cube is expressed in terms of its edgeA How S = 6 A 2 and the volume is V = A 3 . So we can find the edge of the cube and then calculate the surface area:

Thus, the surface area of ​​the cube is:

Answer: 294

27060. The two edges of a cuboid extending from the same vertex are 1 and 2. The surface area of ​​the cuboid is 16. Find its diagonal.

The diagonal of a parallelepiped is calculated by the formula:

where a, b and c are edges.

Let's find the third edge. We can do this using the formula for the surface area of ​​a parallelepiped:

We substitute the data and solve the equation:

Thus, the diagonal will be equal to:

Answer: 3

27063. Find the lateral edge of a regular quadrangular prism if the side of its base is 20 and its surface area is 1760.

At the base of a regular quadrangular prism is a square. It is clear that it is a parallelepiped. The same formulas apply. Let the side edge be equal to x. We can find it using the surface area formula:

Answer: 12

A regular quadrangular prism with a base side of 0.8 and a side edge of 1 is cut from a unit cube. Find the surface area of ​​the remaining part of the cube.

A unit cube is a cube with edge equal to 1.

The surface area of ​​the resulting polyhedron can be calculated as follows: from the surface area of ​​the cube, you need to subtract two areas of the base of the cut out prism and add four areas of the side face of the cut out prism with sides 1 and 0.8:

Answer: 7.92

The area of ​​the face of a rectangular parallelepiped is 48. The edge perpendicular to this face is 8. Find the volume of the parallelepiped.

It is enough to apply the volume formula........................

The volume of a rectangular parallelepiped is equal to the product of its three edges, or the product of the area of ​​the base and the height. In this case, the role of the base is played by the edge, the role of the height is played by the edge, which is perpendicular to it. We get:

Answer: 384

You will solve the following problems without difficulty.

27077. The volume of a rectangular parallelepiped is 64. One of its edges is 4. Find the area of ​​the face of the parallelepiped perpendicular to this edge. Answer: 16.

27078. The volume of a rectangular parallelepiped is 60. The area of ​​one of its faces is 12. Find the edge of the parallelepiped perpendicular to this face. Answer: 5.

27079. Two edges of a rectangular parallelepiped emerging from the same vertex are 8 and 6. The volume of the parallelepiped is 240. Find the third edge of the parallelepiped emerging from the same vertex. Answer: 4.

More for your own solution:

27054. The two edges of a cuboid coming from the same vertex are 3 and 4. The surface area of ​​this cuboid is 94. Find the third edge coming from the same vertex.

P.S: I would be grateful if you tell me about the site on social networks.

Method 1 of 3: Cube the edge of a cube

• Find the length of one edge of the cube. As a rule, the length of a cube edge is given in the problem statement. If you

calculate the volume of a real cubic object, measure its edge with a ruler or tape measure.

Let's consider example. The edge of the cube is 5 cm. Find the volume of the cube.

Cube the length of the edge of the cube. In other words, multiply the length of the cube's edge by itself three times.

If s is the length of the edge of the cube, then

and thus you will calculate cube volume.

This process is similar to the process of finding the area of ​​the base of a cube (equal to the product of the length times

width of the square at the base) and then multiplying the area of ​​the base by the height of the cube (that is,

in other words, you multiply the length by the width by the height). Since in a cube the length of an edge is equal to the width and

equal to the height, then this process can be replaced by raising the edge of the cube to the third power.

In our example cube volume equal to:

• Add volume units to your answer. Since volume is a quantitative

characteristic of the space occupied by a body, then the units of volume measurement are cubic

units (cubic centimeters, cubic meters, etc.).

In our example, the size of the edge of the cube was given in centimeters, so the volume will be measured in cubic

centimeters (or cm 3). So, the volume of the cube is 125 cm3.

If the size of the edge of a cube is given in other units, then the volume of the cube is measured in the corresponding

cubic units.

For example, if the edge of a cube is 5 m (and not 5 cm), then its volume is 125 m 3.

Method 2 of 3: Calculate volume from surface area

• In some problems, the length of the edge of the cube is not given, but other quantities are given with the help of which you

you can find the edge of the cube and its volume. For example, if you are given the surface area of ​​a cube, then divide

it by 6, take the square root from the resulting value and you will find the length of the edge of the cube. Then

Raise the length of the edge of the cube to the third power and calculate the volume of the cube.

Surface area of ​​a cube equal to 6s 2,

Where s - cube edge length(that is, you find the area of ​​one face of the cube and then multiply it by 6, so

like a cube has 6 equal sides).

Let's consider example. The surface area of ​​the cube is 50 cm2. Find the volume of the cube.

• Divide the surface area of ​​the cube by 6 (since the cube has 6 equal sides, you get the area

one face of the cube). In turn, the area of ​​one face of the cube is equal to s 2, Where s- length of the edge of the cube.

In our example: 50/6 = 8.33 cm 2 (remember that area is measured in square units- cm 2,

m 2, etc.).

• Since the area of ​​one face of a cube is s 2, then take the square root of the area value

one face and get the length of the edge of the cube.

In our example, √8.33 = 2.89 cm.

• Cube the resulting value to find the volume of the cube.

In our example: 2.89 * 2.89 * 2.89 = 2.893 = 24.14 cm3. Don't forget to add cubic to your answer.

units.

Method 3 of 3: Calculating Volume Diagonally

• Divide the diagonal of one of the cube's faces by √2 to find the length of the cube's edge. Thus,

if the problem is given the diagonal of a face (any) of a cube, then you can find the length of the edge of the cube by dividing

diagonal by √2.

Let's consider example. The diagonal of the cube's face is 7 cm. Find the volume of the cube. In this case, the length of the cube edge

equal to 7/√2 = 4.96 cm. The volume of the cube is 4.963 = 122.36 cm 3.

Remember: d2 = 2s2,

Where d- diagonal of the cube face, s - edge of the cube. This formula follows from Pythagorean theorem, according to

which the square of the hypotenuse (in our case, the diagonal of the cube face) of a right triangle is equal to

the sum of the squares of the legs (in our case, the edges), that is:

d 2 = s 2 + s 2 = 2s 2.

• Divide the cube's diagonal by √3 to find the length of the cube's edge. Thus, if in the problem

given the diagonal of a cube, then you can find the length of the edge of the cube by dividing the diagonal by √3.

Diagonal of a cube- a segment connecting two vertices that are symmetrical relative to the center of the cube, equal to

D2 = 3s2

(Where D- diagonal of the cube, s- edge of a cube).

This formula follows from the Pythagorean theorem, according to which the square of the hypotenuse (in our case

the diagonal of the cube) of a right triangle is equal to the sum of the squares of the legs (in our case, one leg is

this is an edge, and the second leg is the diagonal of the face of the cube, equal to 2s 2), that is

D 2 = s 2 + 2s 2 = 3s 2.

Let's consider example. The diagonal of the cube is 10 m. Find the volume of the cube.

D2 = 3s2

10 2 = 3s 2

100 = 3s 2

33.33 = s 2

5.77 m = s

The volume of the cube is 5.773 = 192.45 m3.

A cube is one of the simplest three-dimensional objects, both in stereometry and in nature. Before finding the edge of a cube, it is necessary to recall what a cube is. This is a rectangular parallelepiped with equal edges. In addition, the cube is a hexagon whose faces are equal squares. To find the edge of a cube, you need to know some of its parameters - the volume of the cube, the area of ​​the face, the length of the diagonal of the cube or face.

1. In most cases, there are four types of problems that involve an edge of the cube. This is to determine the length of the edge along the diagonal of the cube, along the diagonal of its face, by the volume of the cube and the area of ​​the face. The simplest of them is to find an edge based on the area of ​​the face. After all, the face of a cube is a square with a side that is equal to the edge of the cube. Therefore, the area of ​​this face is equal to the edge of the cube squared. Hence, to find an edge, it is necessary to take the square root of the area of ​​the face. a=vS a – cube edge (length), S – area of ​​one face.
2. It is even easier to find the face of a cube based on its volume, since the volume of the cube will be equal to raising the length of the edge to the 3rd power. Therefore, if we take the cube root (third degree) of the volume, we get the length of the edge a=vV (cube root), here a is the edge of the cube (length), V is its volume.
3. How to find the length of an edge of a cube if the lengths of the diagonals are known. Let us denote: a – cube edge (length), b – cube face diagonal (length), c – cube diagonal (length). The diagonal edges and faces of the cube form an equilateral right triangle. We apply the Pythagorean theorem, where: a^2+a^2=b^2, here (a^ is exponentiation) It turns out: a=v(b^2/2). By taking the square root of half the square of the diagonal of its face, we find the length of the edge of the cube.
4. Find the length of the edge along the diagonal of the cube, where a is the edge of the cube, b is the diagonal of the face, c is the diagonal of the cube. Together they form a right triangle. We proceed from the Pythagorean theorem where: a^2+b^2=c^2. Let's apply the above relationship between the values ​​of a and b and substitute them into the expression b^2=a^2+a^2. Having received: a^2+a^2+a^2=c^2, we find: 3*a^2=c^2, obtaining the final expression - a=v(c^2/3).

If the cube parameters are specified in outdated, national and other specific units, then they should be converted into suitable metric analogues - Cubic Meters, decimeters, centimeters or millimeters.

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