Calculate the area of the parallelepiped online. How to calculate the area of a parallelepiped. What are the types of parallelepipeds
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.
A prism consists of faces, vertices and edges. The distribution of these constituent elements is made in the minimum amount that is necessary to form this geometric shape. The parallelepiped 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, to find out the area of a parallelepiped, it is enough to determine the dimensions of its three faces.
The parallelepiped (in Greek, the term means "parallel faces") has some properties that should be mentioned. Firstly, the symmetry of the figure is confirmed only in the middle of each of its diagonals. Secondly, 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, there are such types of parallelepipeds:
rectangular - consists of rectangular faces;
straight - has only rectangular side faces;
an inclined parallelepiped has lateral faces that are set non-perpendicular to the bases;
cube - consists of square-shaped faces.
Let's try to find the area of a parallelepiped using the example of a 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 \u200b\u200beach face, it is necessary to summarize the results obtained into one number. And to find the area of each of them is not difficult. To do this, you need to multiply the two sides of the rectangle.
A mathematical formula is used to determine the area of a cuboid. It consists of symbolic 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 do an example calculation. Let's say a \u003d 20 cm, b \u003d 16 cm, c \u003d 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 summarized the areas of three faces. Since each face has its own "double", you need to double the resulting value, and we get the area of \u200b\u200bthe 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 parallelepiped 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. Examples can be found in abundance around us. In school curricula on geometry, several lessons are devoted to the study of a parallelepiped. The first of them demonstrate models of a rectangular parallelepiped. Then the students are shown how to inscribe a ball or pyramid, other figures into it, find the area of the parallelepiped. In a word, this is the simplest three-dimensional figure.
A parallelepiped is a polyhedron, which is a particular type of rectangular hexagonal prism. At the base of the parallelepiped lies a rectangle or an equivalent quadrilateral, and parallelograms act as side surfaces. Like any prismatic figure, the box is widespread in real life, but in most cases the real polyhedron takes the form of a rectangular box.
Geometry of the box
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 general, a parallelepiped is a special case of a right quadrangular prism. The parallelepiped is the most common figure in real life. It is the shape of this polyhedron that objects such as houses, rooms, bricks, cardboard boxes, computer blocks, milk packaging, matchboxes and much more have.
The real world consists of various geometric shapes, so you may need a calculator that instantly calculates the surface area of an object that has the shape of a rectangular parallelepiped, whether it is cabinet furniture, a closet or a desktop computer system unit.
Surface area of the parallelepiped
The total surface area of such a prism is defined as the sum of the areas of all faces. The 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 facet.
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 present the formula in terms of the length, width and height of the parallelepiped. In this case, there will be only a slight change in the formula:
S = 2 (ab + bh + ah)
Thus, to determine the total surface area of a prismatic figure, you need to know three of its 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. You may need to calculate the surface area of a prismatic figure in many situations.
Real life examples
wall painting
Let's say you want to paint the walls, floor, and ceiling of the kitchen white. You need to buy enough paint to treat 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 in the form:
Thus, you will need to paint 62 square meters of surface. To do this, you will need to buy 12.4 kg of oil paint or 5 cans of paint of 2.8 kg.
Production
Let's say you're working in a manufacturing environment and you're covering a square steel profile with a protective coating by dipping the parts in a bath of mortar. For the correct calculation of painting parameters, you need to know the surface area of one steel profile, which has the shape of a parallelepiped. The standard square profile has dimensions: length 6 m, side a = 80 mm, side b = 80 mm. For a correct calculation, you need to substitute all dimensions in the same units of measurement, for example, in centimeters. In this case, enter into the online calculator three sides of the box, 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, and rooms, and buildings, and machine parts, and much more. The calculation of the area of this 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.
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 \u200b\u200bthe figure, for which only the lengths of three dimensions of the parallelepiped are needed.
How to find the lateral surface area of a cuboid
It is necessary to distinguish between a rectangular and a right 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 cuboid is calculated using the simple formula P*h, where P is the perimeter and h is the height. The figure shows that the opposite faces 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 cuboid
The total area of the figure consists of the side and the area of 2 bases. How to find the area of a rectangular parallelepiped:
Where a, b and c are the dimensions of the geometric body.
The formulas described are easy to understand and useful in solving many geometry problems. An example of a typical task is shown in the following image.
When solving problems of this kind, it should be remembered that the base of a quadrangular prism is chosen arbitrarily. If we take a face with dimensions x and 3 as the base, then the values of Sside will be different, and Stot will remain 94 cm2.
Cube surface area
A cube is a rectangular parallelepiped with all 3 dimensions equal. In this regard, the formulas for the total and lateral area of a cube differ from the standard ones.
The perimeter of the cube is 4a, therefore, Sside = 4*a*a = 4*a2. These expressions are not required for memorization, but significantly speed up the solution of tasks.
Problem solution example
The above formulas can be used in the search 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.
The parallelepiped is the most common figure that surrounds people. Most of the rooms are just that. It is especially important to know the area of the parallelepiped, at least its side faces, during the repair. After all, you need to know exactly how much material to purchase.
What does he represent?
It is a prism with a square base. Therefore, it has four side faces, which are parallelograms. That is, such a body has only 6 faces.
To determine the 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, only the area of \u200b\u200bthe side faces is also allocated.
What are the types of parallelepipeds?
Inclined. One in which the side faces form an angle other than 90 degrees with the base. Its upper and lower quadrangles do not lie opposite each other, but are shifted.
Straight. A parallelepiped whose side faces are rectangles and whose base is a figure with arbitrary angles.
Rectangular. A special case of the previous view: at its base is a rectangle.
cube. A special type of right box in which all faces are represented by squares.
Some mathematical features of the parallelepiped
There may be a situation 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 by the intersection point into equal parts.
- A more general case, if a segment connects two points on the surface of the body and passes through the intersection point of the diagonals, then it is divided by this point in half.
- For a rectangular parallelepiped, the equality is true, in which in one part of it there is a square of the diagonal, and in the other - the sum of the squares of its height, width and length.
Areas of a right parallelepiped
If we designate the height of the body as “n”, and the perimeter of the base as the letter P os, then the entire lateral surface can be calculated by the formula:
S side \u003d R os * n
Using this formula and determining the area of \u200b\u200bthe base, you can calculate the total area:
S = S side + 2 * S os
In the last entry, S os., that is, the area of \u200b\u200bthe base of the parallelepiped, can be calculated using the formula for a parallelogram. In other words, you need an expression in which you need to multiply the side and the height lowered on it.
Areas of a rectangular parallelepiped
The standard notation 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 \u003d 2 * s * (a + b)
To calculate the total area of a cuboid, you need the following expression:
S = 2 * (av + sun + ac)
If it turns out to be necessary to know the area of \u200b\u200bits base, then it is enough to remember that this is a rectangle, which means that it is enough to multiply “a” and “b”.
cube area
Its lateral surface is formed by four squares. This means that in order to find it, you will 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 is determined by the formula:
S = 6 * a 2
Areas of an inclined parallelepiped
Since its faces are parallelograms, you need to find out the area of \u200b\u200beach of them and then add it up. Fortunately, opposites are equal. Therefore, you need to calculate the areas only three times, and then multiply them by two. If we write this as a formula, we 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 bases.
Related tasks
Task one. Condition. It is necessary to find out the length of the diagonal of a cube if the area of its entire surface is 200 mm 2.
Solution. You need to start by obtaining an expression for the desired value. Its square is equal to three squares of the side of the cube. This means that the diagonal is equal to "a" multiplied by the root of 3.
But the side of the cube is unknown. Here you need to use the fact that the area of \u200b\u200bthe entire surface 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 a cube is 10 mm.
Task two. 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. In it, again, the edge of the body is unknown. But given the volume. 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 \u003d 7 cm.
Answer. S \u003d 294 cm 2.
Task three. Condition. Given a regular quadrangular prism with a base side of 20 dm. It is necessary to find its side edge. It is known that the area of a parallelepiped is 1760 dm 2 .
Solution. You need to start reasoning with the formula for the area of \u200b\u200bthe entire surface of the body. Only in it you need 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 its base is a quadrilateral with equal sides. Hence a \u003d b \u003d 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 desired value "c", which is precisely the side edge of the parallelepiped. To find it, you need to perform transformations:
- divide all inequality by 2;
- then move the terms so that the term 2ac is on the left, and on the right is the area divided by 2 and the square "a", the latter being with the sign "-";
- then divide the equation by 2a.
The result is an expression:
c \u003d (S / 2 - a 2) / (2a)
After substituting all known values and performing actions, it turns out that the side edge is 12 dm.
Answer. The side edge "c" is equal to 12 dm.
Task four. Condition. Given a rectangular parallelepiped. One of its faces has an area equal to 12 cm 2 . It is necessary to calculate the length of the edge that is perpendicular to this face. Additional condition: body volume is 60 cm 3.
Solution. Let the area of the face that is facing the observer be known. If we take as a designation the standard letters for the dimensions of the parallelepiped, then at the base of the rib there will be “a” and “b”, the vertical one will be “c”. Based on this, the area of a known face is determined as the product of "a" by "c".
Now we need to use the known volume. His formula for a cuboid gives the product of all three quantities: "a", "b" and "c". That is, the known area, multiplied by "in", gives the volume. From this 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.
Task five. Condition. Given a right parallelepiped. At its base lies a parallelogram with sides 6 and 8 cm, the acute angle between which is 30º. The side edge has a length of 5 cm. It is required to calculate the total area of the parallelepiped.
Solution. This is the case when you need to find out 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 will require the value of an acute angle. The altitude forms a right 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, and the hypotenuse.
Let the known side of the parallelogram be "a". Then the height will be written as in * sin 30º. Thus, the area of the base is a * b * sin 30º.
With side faces, everything is easier. They are rectangles. Therefore, their areas are the product of one side by the other. The first is a * s, the second is in * s.
It remains to combine everything into one formula and count:
S = 2 * (a * b * sin 30º + a * c + b * c)
After substituting all the values, it turns out that the desired area is 188 cm 2.
Answer. S \u003d 188 cm 2.
In the 5th grade, in the course of mathematics, the topic of a rectangular parallelepiped is studied. Today we will talk about the formulas for finding the area of \u200b\u200ba rectangular parallelepiped of the lateral surface and the area of \u200b\u200bthe full surface of this figure, which most often cause difficulty for students when studying this topic.
Definitions
A parallelepiped is a figure that consists of six quadrilaterals. 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. Capital letters are used to designate the vertices of the figure.
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 sides of the cuboid are equal.
The sides of the faces are edges, the figure has 12 edges. The length of the edges determines the main characteristics of a rectangular parallelepiped: area, perimeter, volume.
Rice. 1. Rectangular box
We often see examples of such figures in our lives: a brick, a box, a 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 separately the area of each lateral face, and then sum the resulting values.
$S = ac, a, b, c$ are the sides of the figure.
Rice. 2. Rectangular box
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, in order to calculate the total surface area of a rectangular parallelepiped, it is necessary to add the side surface area and two base areas. The result is 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 sign of the area, for example, S p.p - the area of \u200b\u200bthe full surface, or S b.p - the area of \u200b\u200bthe lateral surface. This helps during the execution of the task not to confuse the necessary data.
Task example
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.
