All the subtleties of how to calculate the area of the parallelepiped
The parallelepiped is the most common shape among those around people. Most of the premises are his. 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 is he like?
This is a prism with a quadrangular base. Therefore, it has four side faces that are parallelograms. That is, such a body has only 6 faces.
To determine the parallelepiped in space, it determines the area and volume. The first one can be both separately for each face and for the whole surface. In addition, emit more and only the side faces.
What are the types of parallelepipeds?
Inclined. One in which the lateral faces form with the base an angle different from 90 degrees. Its upper and lower quadrilaterals do not lie opposite each other, but are shifted.
Straight.A parallelepiped, the side faces of which are rectangles, and at the base lies a figure with arbitrary angles.
Rectangular. A special case of the previous type: in its base is a rectangle.
Cube A special type of a straight parallelepiped in which all faces are represented by squares.
Some mathematical features of the parallelepiped
A situation may arise when they are useful in finding the area of the parallelepiped.
- The edges that lie opposite each other are not only parallel, but equal.
- The diagonal of the parallelepiped intersection point is divided into equal parts.
- More generally, if a segment connects two points on the surface of a body and passes through the intersection point of the diagonals, then it is divided in half by this point.
- For a rectangular parallelepiped, the equality is valid, in which in one part there is a square of a diagonal, and in the other - the sum of squares of its height, width and length.
If we denote the height of the body as "n", and the base perimeter is Pwaspthen the entire side surface can be calculated by the formula:
Sside= Pwasp* n
Using this formula and determining the area of the base, we can count the total area:
S = Sside+ 2 * Swasp
In the last entry Swasp., that is, the area of the base of the parallelepiped, can be calculated by the formula for the parallelogram. In other words, you need an expression in which you need to multiply the side and the height lowered onto it.
The standard designation of the length, width and height of such a body with the letters “a”, “b” and “c”, respectively, is adopted. The area of the side surface will be expressed by the formula:
Sside= 2 * s * (a + c)
To calculate the total area of a rectangular parallelepiped, we need the following expression:
S = 2 * (aw + bs + as)
If it turns out to be necessary to know the area of its base, then it suffices to recall that this is a rectangle, which means that it is enough to multiply "a" and "c".
Its side surface is formed by four squares. So, to find it, you need to use the well-known formula for the square and multiply it by four.
Sside= 4 * a2
And due to the fact that its bases are the same squares, the total area is determined by the formula:
S = 6 * a2
Slanted Parallelepiped Squares
Since its faces are parallelograms, you need to find out the area of each of them and then fold it. Fortunately, the opposite are equal. Therefore, it is necessary to calculate the area only three times, and then multiply them by two. If you write this in the form of a formula, you get the following:
Sside= (S1+ S2) * 2,
S = (S1+ S2+ S3) * 2
Here s1and s2are the areas of the two side faces, and S3- grounds.
The first task.Condition.You need to know the length of the diagonal of the cube, if the area of its entire surface is 200 mm2.
Decision.We need to start by getting 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 will 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 particular S and 6.
It remains only to count. The edge of the cube is equal to √ (200/6), which is equal to 10 / √3 (mm). Then the diagonal will be equal to (10 / √3) * √3 = 10 (mm).
Answer.The cube diagonal is 10 mm.
The second task. Condition. It is necessary to calculate the surface area of the cube, if it is known that its volume is 343 cm2.
Decision.You will need to use the same formula for the cube area. Again, it is again unknown edge of the body. But given the volume. From the formula for the cube is very easy to learn "a". It will be equal to the cube root of 343. A simple calculation gives the value for the edge: a = 7 cm.
Now it remains only to count its square and multiply by 6. a2= 72= 49, hence the area will be equal to 49 * 6 = 294 (cm2).
Answer.S = 294 cm2.
The third task.Condition. A regular quadrangular prism with a base of 20 dm is given. It is necessary to find its side edge. It is known that the parallelepiped area is equal to 1760 dm2.
Decision.Begin 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 “c” are equal. This follows from the statement that the prism is correct. Hence, at its base lies a quadrilateral with equal sides. From here a = v = 20 dm.
Given this circumstance, the area formula will be simplified to this:
S = 2 * (a2+ 2as).
Everything is known in it, except for the sought-after quantity “c”, which is precisely the side edge of the parallelepiped. To find it, you need to perform the conversion:
- divide all inequality by 2;
- then move the terms so that the left side is 2as, and on the right is the area divided by 2 and the square “a”, the latter being marked with “-”;
- then divide the equality by 2a.
The result is the expression:
c = (s / 2 - a2) / (2a)
After substituting all known values and performing actions, it turns out that the side edge is equal to 12 dm.
Answer. The side edge “c” is 12 dm.
The fourth task.Condition.Given a rectangular parallelepiped. One of its faces has an area of 12 cm.2. It is necessary to calculate the length of the edge, which is perpendicular to this face. Additional condition: body volume is 60 cm3.
Decision.Let the area of that face, which is located facing the observer, be known. If the standard letters for parallelepiped dimensions are taken for designation, then at the base of the edge there will be “a” and “b”, the vertical one will be “c”. Based on this, the area of a known face is defined as the product “a” on “c”.
Now you need to use a known volume. His formula for a rectangular parallelepiped gives the product of all three quantities: "a", "in" 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:
12 * = 60.
Elementary calculation gives the result 5.
Answer.The desired edge is 5 cm.
The fifth task.Condition.Given a straight parallelepiped.At its base is a parallelogram with sides of 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.
Decision.This is the case when you need to know the area of all faces separately. Or, more precisely, three pairs: the base and the two side.
Since a parallelogram is located 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 must be counted. This will require a sharp angle value. The height forms a right triangle in the 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 famous side of the parallelogram is “a”. Then the height will be written as * sin 30º. Thus, the base area is a * c * sin 30º.
With side edges, everything is easier. They are rectangles. Therefore, their area is the product of one side to the other. The first - a * s, the second - in * s.
It remains to combine all in one formula and count:
S = 2 * (a * c * sin 30º + a * s + b * s)
After substitution of all quantities, it turns out that the required area is 188 cm2.
Answer.S = 188 cm2.