Parallelepiped, cube. Detailed theory with examples

The prism is called parallelepiped if its bases are parallelograms. Cm. Fig.1.

Properties of the box:

The opposite faces of the parallelepiped are parallel (i.e. lie in parallel planes) and equal.

The diagonals of the parallelepiped intersect at one point and bisect that point.

Adjacent faces of a box are two faces that have a common edge.

Opposite faces of a parallelepiped– faces that do not have common edges.

Opposite vertices of the box are two vertices that do not belong to the same face.

Diagonal of the box A line segment that connects opposite vertices.

If the lateral edges are perpendicular to the planes of the bases, then the parallelepiped is called direct.

A right parallelepiped whose bases are rectangles is called rectangular. A prism all of whose faces are squares is called cube.

Parallelepiped A prism whose bases are parallelograms.

Right parallelepiped- a parallelepiped whose lateral edges are perpendicular to the plane of the base.

cuboid is a right parallelepiped whose bases are rectangles.

Cube is a rectangular parallelepiped with equal edges.

Parallelepiped a prism is called, the base of which is a parallelogram; thus, the parallelepiped has six faces and all of them are parallelograms.

Opposite faces are pairwise equal and parallel. The parallelepiped has four diagonals; they all intersect at one point and divide in half at it. Any face can be taken as a base; the volume is equal to the product of the base area and the height: V = Sh.

A parallelepiped whose four lateral faces are rectangles is called a right parallelepiped.

A right parallelepiped, in which all six faces are rectangles, is called rectangular. Cm. Fig.2.

The volume (V) of a right parallelepiped is equal to the product of the base area (S) and the height (h): V = Sh .

For a rectangular parallelepiped, in addition, the formula V=abc, where a,b,c are edges.

The diagonal (d) of a cuboid is related to its edges by the relationship d 2 \u003d a 2 + b 2 + c 2 .

cuboid- a parallelepiped whose lateral edges are perpendicular to the bases, and the bases are rectangles.

Properties of a cuboid:

In a cuboid, all six faces are rectangles.

All dihedral angles of a cuboid are right angles.

The square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions (lengths of three edges that have a common vertex).

The diagonals of a rectangular parallelepiped are equal.

A rectangular parallelepiped, all of whose faces are squares, is called a cube. All edges of a cube are equal; the volume (V) of a cube is expressed by the formula V=a 3, where a is the edge of the cube.

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or (equivalently) a polyhedron with six faces that are parallelograms. Hexagon.

The parallelograms that make up the parallelepiped are faces this parallelepiped, the sides of these parallelograms are parallelepiped edges, and the vertices of the parallelograms are peaks parallelepiped. Each face of a parallelepiped is parallelogram.

As a rule, any 2nd opposite faces are distinguished and called them the bases of the parallelepiped, and the remaining faces side faces of the parallelepiped. The edges of the parallelepiped that do not belong to the bases are side ribs.

The 2 faces of a cuboid that share an edge are related, and those that do not have common edges - opposite.

A segment that connects 2 vertices that do not belong to the 1st face is the diagonal of the parallelepiped.

The lengths of the edges of a cuboid that are not parallel are linear dimensions (measurements) a parallelepiped. A rectangular parallelepiped has 3 linear dimensions.

Types of parallelepiped.

There are several types of parallelepipeds:

Direct is a parallelepiped with an edge perpendicular to the plane of the base.

A cuboid with all 3 dimensions equal in magnitude is cube. Each of the faces of the cube is equal squares.

Arbitrary parallelepiped. The volume and ratios in a skew box are mostly defined using vector algebra. Volume of the box equals absolute value mixed product of 3 vectors, which are defined by 3 sides of the parallelepiped (which come from the same vertex). The ratio between the lengths of the sides of the parallelepiped and the angles between them shows the statement that the Gram determinant of the given 3 vectors is equal to the square of their mixed product.

Properties of a parallelepiped.

• The parallelepiped is symmetrical about the midpoint of its diagonal.
• Any segment with ends that belong to the surface of the parallelepiped and which passes through the midpoint of its diagonal is divided by it into two equal parts. All diagonals of the parallelepiped intersect at the 1st point and are divided by it into two equal parts.
• Opposite faces of a parallelepiped are parallel and have equal dimensions.
• The square of the length of the diagonal of a cuboid is