We calculate the area of the parallelepiped. How to calculate the area of a parallelepiped How is the area of a rectangular parallelepiped measured?

The box is the most common shape that surrounds people. Most of the premises are just him. It is especially important to know the area of the parallelepiped, at least of its lateral faces, during the repair. After all, you need to know exactly how much material to buy.

What is he like?

It is a prism with a quadrangular base. Therefore, it has four side faces, which are parallelograms. That is, such a body has only 6 faces.

To define a parallelepiped in space, its area and volume are determined. The first can be either separately for each face, or for the entire surface. In addition, the area of only the lateral faces is also distinguished.

What are the types of parallelepipeds?

Inclined. One in which the side faces form an angle with the base that is different from 90 degrees. Its upper and lower quadrangles do not lie opposite each other, but are shifted.

Straight. A parallelepiped, the side faces of which are rectangles, and at the base there is a figure with arbitrary angles.

Rectangular. A special case of the previous type: there is a rectangle at its base.

Cube A special type of straight parallelepiped in which all faces are represented by squares.

Some mathematical features of a parallelepiped

A situation may arise where they are useful in finding the area of a parallelepiped.

The faces that lie opposite each other are not only parallel, but also equal.
The diagonals of the parallelepiped are divided into equal parts by the intersection point.
A more general case, if a segment connects two points on the surface of the body and passes through the point of intersection of the diagonals, then it is divided by this point in half.
For a rectangular parallelepiped, equality is true, in which in one part there is a square of the diagonal, and in the other - the sum of the squares of its height, width and length.

Squares of a straight parallelepiped

If we designate the height of the body as "n", and the perimeter of the base with the letter P os, then the entire lateral surface can be calculated by the formula:

S side = P o * n

Using this formula and determining the area of the base, you can calculate the total area:

S = S side + 2 * S os

In the last record, S os., That is, the area of the base of the parallelepiped, can be calculated using the formula for the parallelogram. In other words, you need an expression in which you need to multiply the side and the height dropped by it.

Areas of a rectangular parallelepiped

The standard designation for the length, width and height of such a body is adopted by the letters "a", "b" and "c", respectively. The lateral surface area will be expressed by the formula:

S side = 2 * c * (a + b)

To calculate the total area of a rectangular parallelepiped, you need the following expression:

S = 2 * (ab + sun + ac)

If it turns out to be necessary to know the area of its base, then it is enough to remember that it is a rectangle, which means that it is enough to multiply "a" and "b".

Cube areas

Its lateral surface is formed by four squares. So, to find it, you need to use the formula known for the square and multiply it by four.

S side = 4 * a 2

And due to the fact that its bases are the same squares, the total area will be determined by the formula:

S = 6 * a 2

Oblique parallelepiped areas

Since its faces are parallelograms, you need to find out the area of each of them and then add it up. Fortunately, the opposites are equal. Therefore, you only need to calculate the areas three times, and then multiply them by two. If you write this in the form of a formula, you get the following:

S side = (S 1 + S 2) * 2,

S = (S 1 + S 2 + S 3) * 2

Here S 1 and S 2 are the areas of the two side faces, and S 3 are the base.

Tasks by topic

The first task. Condition. It is necessary to find out the length of the diagonal of the cube if the area of its entire surface is 200 mm 2.

Solution. You need to start by getting an expression for the desired value. Its square is equal to three squares of the side of a cube. This means that the diagonal is equal to "a" times the root of 3.

But the side of the cube is unknown. Here you need to take advantage of the fact that the entire surface area is known. From the formula it turns out that "a" is equal to the square root of the quotient S and 6.

Answer. The diagonal of the cube is 10 mm.

Second task. Condition. It is necessary to calculate the surface area of a cube if it is known that its volume is 343 cm 2.

Solution. You will need to use the same formula for the area of a cube. The rib of the body is again unknown in it. But the volume is given. From the formula for the cube it is very easy to find out "a". It will be equal to the cube root of 343. A simple calculation gives this value for the edge: a = 7 cm.

Answer. S = 294 cm 2.

The third task. Condition... A regular quadrangular prism with a base side of 20 dm is given. It is necessary to find its lateral rib. It is known that the area of a parallelepiped is 1760 dm 2.

Solution. You need to start the reasoning with the formula for the area of the entire surface of the body. Only in it it is necessary to take into account that the edges "a" and "b" are equal. This follows from the statement that the prism is correct. This means that at its base lies a quadrangle with equal sides. Hence a = b = 20 dm.

Given this circumstance, the area formula will be simplified to the following:

S = 2 * (a 2 + 2ac).

Everything is known in it, except for the sought-for value "c", which is precisely the lateral edge of the parallelepiped. To find it, you need to perform transformations:

divide all inequalities by 2;
then transfer the terms so that the term 2ac is on the left, and the area and the square "a" divided by 2, on the right, and the latter will be with the "-" sign;
then divide the equality by 2a.

As a result, you get the expression:

c = (S / 2 - a 2) / (2a)

After substituting all known values and performing the actions, it turns out that the side edge is 12 dm.

Answer... The side rib "c" is 12 in.

The fourth task. Condition. You are given a rectangular parallelepiped. One of its faces has an area of 12 cm 2. You need to calculate the length of the edge that is perpendicular to this face. Additional condition: the volume of the body is 60 cm 3.

Solution. Let the area of the face that is facing the observer be known. If we take the standard letters for the dimensions of a parallelepiped as a designation, then at the base of the edges there will be "a" and "b", vertical - "c". Based on this, the area of a known facet will be determined as the product of "a" and "c".

Now you need to use the known volume. Its formula for a rectangular parallelepiped gives the product of all three quantities: "a", "b" and "c". That is, the known area multiplied by "in" gives the volume. Hence, it turns out that the desired edge can be calculated from the equation:

An elementary calculation gives the result 5.

Answer. The desired edge is 5 cm.

Fifth task. Condition. You are given a straight parallelepiped. At its base lies a parallelogram with sides of 6 and 8 cm, the acute angle between which is 30º. The side edge is 5 cm long. You want to calculate the total area of the parallelepiped.

Solution. This is the case when you need to know the areas of all faces separately. Or, more precisely, three pairs: a base and two side ones.

Since there is a parallelogram at the base, its area is calculated as the product of the side and the height to it. The side is known, but the height is not. It needs to be counted. This requires an acute angle value. The height forms a right-angled triangle in a parallelogram. In it, the leg is equal to the product of the sine of the acute angle, which is opposite to it, by the hypotenuse.

Let the known side of the parallelogram be "a". Then the height will be recorded as * sin 30º. Thus, the area of the base is equal to a * b * sin 30º.

With the side edges, everything is easier. They are rectangles. Therefore, their areas are the product of one side to the other. The first is a * s, the second is b * s.

It remains to combine everything into one formula and count:

S = 2 * (a * b * sin 30º + a * c + b * c)

After substitution of all values, it turns out that the required area is 188 cm 2.

Answer. S = 188 cm 2.

In the 5th grade, in the mathematics course, the topic of a rectangular parallelepiped is studied. Today we will talk about the formulas for finding the area of a rectangular parallelepiped of the lateral surface and the area of the full surface of this figure, which most often cause difficulty for students when studying this topic.

Definitions

A box is a shape that consists of six quadrangles. If there is a rectangle at the base of this figure, then the polyhedron is called a rectangular parallelepiped.

A rectangular parallelepiped has four side faces. Two of them are called the base of the polyhedron. To designate the tops of the figure, large Latin letters are used.

If two faces do not have a common edge, then they are called opposite. Since each face is a rectangle, where the opposite sides are equal, then the opposite faces of the rectangular parallelepiped are equal.

The sides of the faces are edges, and the shape has 12 edges. The length of the edges determines the main characteristics of a rectangular parallelepiped: area, perimeter, volume.

Rice. 1. Rectangular parallelepiped

We often see examples of such figures in our life: brick, box, computer system unit.

A mathematical figure - a rectangular parallelepiped is actively used in art, architecture and other fields.

There are several types of parallelepipeds, with a base in the form of a square, parallelogram or rectangle.

Formula for finding area

In order to find the area of the lateral surface of a rectangular parallelepiped, it is necessary to calculate the area of each lateral face separately, and then sum the resulting values.

$ S = ac, a, b, c $ - the sides of the figure.

Rice. 2. Rectangular parallelepiped

And since the opposite faces are equal, that is, $ AMPD = BNKC $, $ AMNB = DPKC $, their sum will be the area of the lateral surface of the polygon.

Accordingly, to calculate the total surface area of a rectangular parallelepiped, it is necessary to add the lateral surface area and the two base areas. As a result, you get the formula for the area of a rectangular parallelepiped.

$ S = 2 (ab + ac) + 2 bc = 2 (ab + ac + bc) $

Sometimes, for clarification, a short designation is written near the area sign, for example, S p.p - the total surface area, or S b.p - the lateral surface area. This helps not to confuse the required data during the execution of the task.

Example task

Find the total surface area of a rectangular parallelepiped if the length and width of the base are 4 cm and 3 cm, respectively, and the height is 2 cm.

A parallelepiped is a polyhedron, which is a particular view of a rectangular hexagonal prism. At the base of the parallelepiped lies a rectangle or an equivalent quadrangle, and parallelograms act as side surfaces. Like any prismatic figure, a parallelepiped is widespread in real life, but in most cases, a real polyhedron takes the shape of a rectangular parallelepiped.

Parallelepiped geometry

A rectangular parallelepiped consists of two identical rectangles lying in parallel planes and four rectangles connecting them, which form the side surface of the figure. In the general case, a parallelepiped is a special case of a straight quadrangular prism. The box is the most common shape in real life. It is the shape of this polyhedron that has such objects as houses, rooms, bricks, cardboard boxes, computer blocks, milk cartons, match boxes and much more.

The real world consists of various geometric shapes, so you may need a calculator that instantly calculates the surface area of an object in the shape of a rectangular parallelepiped, be it cabinet furniture, a closet or a desktop computer system unit.

Surface area of a parallelepiped

The total surface area of such a prism is defined as the sum of the areas of all faces. A parallelepiped is a hexagon, each pair of faces of which are equal to each other. This means that each face of the parallelepiped has its own congruent pair. Thus, the surface area of a given prismatic figure is expressed as twice the sum of the areas of each face.

S = 2 (Sa + Sb + Sc)

Since each face of a parallelepiped is a regular rectangle, the area of one face is defined as the product of the sides of the polygon. If a prismatic figure has sides a, b and c, then its total surface area will be equal to:

S = 2 (ab + bc + ac)

For a simpler understanding, we can represent the formula in terms of the length, width and height of the parallelepiped. In this case, there will be only a small change in the formula:

S = 2 (ab + bh + ah)

Thus, to determine the total surface area of a prismatic figure, you need to know its three parameters. Enter this data into the online calculator form and you will get an instant result. In addition, the calculator will immediately calculate the length of the diagonal of the polyhedron. Calculating the surface area of a prismatic figure can be useful in many situations.

Examples from life

Painting the walls

Let's say you want to paint the walls, floor and ceiling of your kitchen with white paint. You need to buy enough paint to paint the selected room. Knowing that the consumption of oil paint per 1 square meter of surface is approximately 200 grams, you can determine how much material you need to work. Let the kitchen space be 3 m high, 2 m wide and 5 m long.Enter this data into the online calculator and you will get the result as:

Thus, you will need to paint 62 square meters of the surface. To do this, you will need to buy 12.4 kg of oil paint or 5 cans of paint of 2.8 kg each.

Production

Let's say you are in a manufacturing facility and you are covering a steel square profile with a protective coating by dipping the parts into a bathroom with a solution. For the correct calculation of the painting parameters, you need to know the surface area of one steel profile, which has the shape of a parallelepiped. A standard square profile has dimensions: length 6 m, side a = 80 mm, side b = 80 mm. For the correct calculation, you need to substitute all dimensions in the same units of measurement, for example, in centimeters. In this case, drive three sides of the parallelepiped into the online calculator, which are equal to 600, 8 and 8. You will get the result in the form:

Thus, the total surface area of the steel profile is 19,328 square centimeters or 1.9828 square meters. Knowing the surface area of one profile, you can easily determine the parameters for painting parts with a protective coating.

Conclusion

A large number of real objects have the shape of a parallelepiped: these are bricks, rooms, buildings, machine parts, and much more. Calculation of the area of a given polyhedron may be needed in the most unexpected situations, such as everyday problems or professional calculations. Our online calculator will help you quickly determine the volumes and surface areas of any regular geometric shapes.

One of the simplest can be called a parallelepiped. It has the shape of a prism, at the base of which is a parallelogram. It is not difficult to calculate the area of a parallelepiped, since the formula is very simple.

The prism is made up of faces, vertices and edges. The distribution of these constituent elements is performed in the minimum amount that is necessary to form this geometric shape. The box contains 6 faces, which are connected by 8 vertices and 12 edges. Moreover, the opposite sides of the parallelepiped will always be equal to each other. Therefore, in order to find out the area of a parallelepiped, it is enough to determine the dimensions of its three faces.

The parallelepiped (translated from Greek, the term means "parallel faces") has some properties that should be mentioned. First, the symmetry of the figure is confirmed only in the middle of each of its diagonal. Second, by drawing a diagonal between any of the opposite vertices, you can find that all vertices have a single intersection point. It is also worth noting the property that opposite faces are always equal and will necessarily be parallel to each other.

In nature, the following types of parallelepipeds are distinguished:

rectangular - consists of rectangular faces;

straight - has only rectangular side faces;

the inclined parallelepiped contains side faces that are not perpendicular to the bases;

cube - consists of square-shaped faces.

Let's try to find the area of a parallelepiped using the example of the rectangular type of this figure. As we already know, all its faces are rectangular. And since the number of these elements is reduced to six, then, having learned the area of each face, you need to sum the resulting results into one number. And finding the area of each of them will not be difficult. To do this, you need to multiply the two sides of the rectangle.

Uses a mathematical formula to determine the area of a rectangular box. It consists of sign symbols denoting faces, area, and looks like this: S = 2 (ab + bc + ac), where S is the area of the figure, a, b are the sides of the base, c is the side edge.

Let's give an approximate calculation. Let's say a = 20 cm, b = 16 cm, c = 10 cm. Now you need to multiply the numbers in accordance with the requirements of the formula: 20 * 16 + 16 * 10 + 20 * 10 and we get the number 680 cm2. But this will be only half of the figure, since we have learned and summed up the areas of the three faces. Since each face has its own "twin", you need to double the resulting value, and we get the area of a parallelepiped equal to 1360 cm2.

To calculate the lateral surface area, apply the formula S = 2c (a + b). The area of the base of a box can be found by multiplying the lengths of the sides of the base by each other.

In everyday life, parallelepipeds can often be found. The shape of a brick, a wooden box of an ordinary matchbox, reminds us of their existence. Everyone can find examples in abundance around us. In school geometry programs, several lessons are allocated for the study of a parallelepiped. The first of them show models of a rectangular parallelepiped. Then the students are shown how to inscribe a ball or pyramid, other shapes, and find the area of a parallelepiped. In short, this is the simplest three-dimensional figure.

A parallelepiped is a quadrangular prism with a parallelogram at its base. There are ready-made formulas for calculating the lateral and total surface area of a figure, for which only the lengths of three dimensions of a parallelepiped are needed.

How to find the side area of a rectangular parallelepiped

It is necessary to distinguish between rectangular and straight parallelepiped. The base of a straight figure can be any parallelogram. The area of such a figure must be calculated using other formulas.

The sum S of the side faces of a rectangular parallelepiped is calculated using the simple formula P * h, where P is the perimeter and h is the height. The figure shows that the opposite sides of a rectangular parallelepiped are equal, and the height h coincides with the length of the edges perpendicular to the base.

Surface area of a rectangular parallelepiped

The total area of the figure consists of the side and the area of 2 bases. How to find the areas of a rectangular parallelepiped:

Where a, b and c are the dimensions of the geometric body.
The described formulas are easy to understand and useful in solving many geometry problems. An example of a typical job is shown in the following image.

When solving such problems, it should be remembered that the base of the quadrangular prism is chosen arbitrarily. If we take the edge with measurements x and 3 as the base, then the S side values will be different, and S total will remain 94 cm2.

Cube surface area

A cube is a rectangular parallelepiped in which all 3 dimensions are equal. In this regard, the formulas for the total and lateral areas of the cube differ from the standard ones.

The perimeter of the cube is 4a, therefore, Side = 4 * a * a = 4 * a2. These expressions are not required for memorization, but they significantly speed up the solution of tasks.

An example of solving the problem

The above formulas can be used when searching for the diagonals of a parallelepiped.

To find B1D, it is enough to apply the Pythagorean theorem: the sum of the squares of the legs is equal to the square of the hypotenuse.

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We calculate the area of ​​the parallelepiped. How to calculate the area of ​​a parallelepiped How is the area of ​​a rectangular parallelepiped measured?