# Advice 1: How to find the cross section of the parallelepiped

Cross section of geometric figures of various shapes. The parallelepiped section is always a rectangle or square. It has a number of options that can be found analytically.
Instruction
1
Using a box can hold four sections, which are squares or rectangles. He has two diagonal and two cross-sections. They usually have different sizes. The exception is the cube, which are the same.
Before you build section of the box, make a representation that represents this figure. There are two types of parallelepipeds - normal and rectangular. Conventional parallelepiped facets are at an angle to the base, they are rectangular and perpendicular to it. All faces of a rectangular parallelepiped are rectangles or squares. It follows from this that a cube is a special case of a rectangular parallelepiped.
2
Any section of a parallelepiped has certain characteristics. The main ones are area, perimeter, length of the diagonals. If the conditions of the problem known to the parties section, or any other parameters, this is enough to find its perimeter or area. The parties determined also diagonal sections. The first of these parameters - the area of the diagonal section.
To find the area of the diagonal section, you need to know the height and base side of the parallelepiped. If the length and width of the base of the box, then find the diagonal by using the Pythagorean theorem:
d=√a^2+b^2.
Finding the diagonal and knowing the height of a cuboid, calculate the cross-sectional area of a parallelepiped:
S=d*h.
3
The perimeter of a diagonal cross-section is also possible to calculate two values - the diagonal of the base and the height of the parallelepiped. In this case, first find the two diagonals (upper and lower grounds) by the Pythagorean theorem, and then fold over with double height.
4
If we draw a plane parallel to the edges of the parallelepiped, it is possible to obtain the cross-section is rectangle, the sides of which are one side of the base of the box and height. The area of this section find the following:
S=a*h.
The perimeter of this section find in the same way according to the following formula:
p=2*(a+h).
5
The latter case occurs when the section passes in parallel to the two bases of the parallelepiped. Then its area and perimeter equal to area value and the perimeter of the grounds, ie:
S=a*b - cross-sectional area;

p=2*(a+b).

# Advice 2: How to find the diagonal of a parallelepiped

The parallelepiped is a particular case of the prism, in which all six faces are parallelograms or rectangles. A parallelepiped with rectangular faces is called a rectangular. Of parallelepiped has four diagonals intersect. If the three edges a, b, C, to find all the diagonals of a rectangular parallelepiped, you can do additional build.
Instruction
1
Draw a rectangular parallelepiped. Write known data: three edges a, b, C. First, build a single diagonal m. For its determination we use the property of a rectangular parallelepiped, according to which all the corners are straight.
2
Construct a diagonal n one of the faces of the parallelepiped. Build a guide so that the edge is known, the required diagonal of the parallelepiped and the diagonal faces together formed a right triangle a, n, m.
3
Find built diagonal faces. It is the hypotenuse of another right triangle b, C, n. According to the Pythagorean theorem n2 = S2 + b2. Calculate this expression and take the square root of the obtained values, it will be the diagonal of n faces.
4
Find the diagonal of a parallelepiped m. To do this in a right triangle a, n, m find the unknown hypotenuse: m2 = n2 + a2. Substitute known values, then calculate the square root. The result is the diagonal of a parallelepiped m.
5
Likewise consistently spend the remaining three diagonals of a parallelepiped. Also, for each of them will complete additional construction of the diagonals of the adjacent faces. Formed by considering right triangles and applying the Pythagorean theorem, find the values of the remaining diagonals of a rectangular parallelepiped.

# Advice 3: How to find the volume of a parallelepiped

The form of a parallelepiped have a real objects. Examples are the room and the pool. Details with this form - are not uncommon in the industry. For this reason, there is often the problem of finding the volume of this shape.
Instruction
1
A parallelepiped is a prism whose base is a parallelogram. The parallelepiped has a face - all the planes that form the given shape. In total he has six faces and all are parallelograms. Opposing faces are equal and parallel. In addition, it has diagonals that intersect at one point and it split in half.
2
The parallelepiped is of two types. At first all faces are parallelograms, and the second with rectangles. The last of them is called a rectangular parallelepiped. He has all faces rectangular, and the side faces are perpendicular to the base. If the cuboid has faces, the Foundation of which the squares, it is called a cube. In this case, its faces and edges are equal. An edge is called a face of any polyhedron, which include the box.
3
In order to find the volume of a parallelepiped, you must know the area of its base and height. The amount is based on what appears parallelepiped in terms of the problem. The common box in the base is a parallelogram, and rectangular - rectangular or square, whose corners are always straight. If the base of a parallelepiped is the parallelogram, then its volume is as follows:
V=S*H, where S is the total area of the base H is the height of the box
The height of the parallelepiped is usually its lateral edge. At the base of the parallelepiped may also be a parallelogram, not a rectangle. Of course of plane geometry it is known that the area of a parallelogram is equal to:
S=a*h, where h is the height of a parallelogram, a is the length of the base, ie :
V=a*hp*H
4
If the second case is when the base of the box - a rectangle, the volume is calculated using the same formula, but the footprint is in a little different way:
V=S*H,
S=a*b, where a and b are, respectively, side of the rectangle and edges of the parallelepiped.
V=a*b*H
5
For finding the volume of a cube should be guided by simple logical ways. Since all faces and edges of a cube are equal, and at the base of the cube - square, in accordance with formulas specified above, we can derive the following formula:
V=a^3

# Advice 4: How to build a cross-section of a parallelepiped

In many textbooks there are jobs associated with the construction of sections of various geometric shapes including parallelepipeds. To cope with this task, you should be armed with some knowledge.
You will need
• paper;
• - handle;
• - the range.
Instruction
1
On a sheet of paper, draw a box. If your task says that the box needs to be rectangular, make it square corners. Remember that opposite edges must be parallel to each other. Name its vertices, for example, S1, T1, T, R, P, R1, P1 (as shown in the figure).
2
On the verge SS1TT1 put 2 points: A and C, let point a be on the segment S1T1 and a point on the segment S1S. If your task doesn't say where it should be these points, and do not specify the distance from the vertices, put them arbitrarily. Draw a straight line through points A and C. Continue this line to its intersection with the line segment ST. Label the point of intersection, let it be a point of M.
3
Put a point on the segment RT, label it as point B. draw a straight line through the points M and B. the intersection of this line with the edge of the label SP as point K.
4
Connect the points K and C. They must lie on the same face PP1SS1. Then through point B draw a straight line, parallel to the segment of the COP, will continue the line to the intersection with an edge of R1T1. The intersection of the label as point E.
5
Connect the points A and E. then select the resulting polygon ACKBE a different color – this will be a cross section of a given parallelepiped.
Note
Remember that when you build a cross-section of the parallelepiped is possible to connect only those points that lie in the same plane, if you had enough points for the build section, add them by extending the line segments to the intersection with the face on which the desired point.
Only the parallelepiped can be constructed of 4 sections: 2 diagonal and 2 transverse. For clarity, select the resulting polygon cross section, this can just stroke or stroke it in a different color.

# Advice 5: How to find the area of the diagonal cross section of the prism

Prism — a polyhedron with two parallel bases and lateral faces in the shape of a parallelogram and in an amount equal to the number of sides of the base polygon.
Instruction
1
In any prism the lateral edges are at an angle to the plane of the base. A special case is a straight prism. Her sides lie in planes perpendicular to the bases. In straight prism lateral faces rectangles and the side edges equal to the height of the prism.
2
The diagonal section of the prism — part of the plane completely enclosed in the inner space of the polyhedron. Diagonal cross section can be limited by two side edges and a geometric body diagonals of the bases. It is obvious that the number of possible diagonal sections is determined by the number of diagonals in a polygon base.
3
Or borders of a diagonal cross-section can serve as the diagonal side faces and the opposite side of the bases of the prism. Diagonal cross section of a rectangular prism is a rectangle. In the General case of an arbitrary prism form a diagonal of the cross section is a parallelogram.
4
In a rectangular prism of square cross-section diagonal of S is defined by the formula:
S=d*H
where d is the diagonal of the base,
H — the height of the prism.
Or S=a*D
where a — side of the base, at the same time belonging to the cutting plane
D — the diagonal of the side face.
5
In the random indirect prism diagonal cross section is a parallelogram, one side of which is equal to a lateral edge of the prism, the other diagonal of the base. Or sides, diagonal lines can be diagonal side faces and the sides of the bases between the vertices of the prism, where the drawn diagonal side surfaces. The area of a parallelogram S is determined by the formula:
S=d*h
where d is the diagonal of the base of the prism,
h — the height of a parallelogram — diagonal cross-section of the prism.
Or S=a*h
where a — side of the base of the prism, which is the border of the diagonal lines,
h — the height of the parallelogram.
6
To determine the height of the diagonal section is not enough to know the linear dimensions of the prism. The necessary data about the inclination of the prism to the plane of the base. A further problem is reduced to the consecutive solution of several triangles depending on the source of data on the angles between the elements of prism.
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