Parallelepiped ℹ️ Definition, properties, species, formulas for calculating area, volume and perimeter of the geometric shape, proof of theorems

Parallelepiped

general characteristics

There are many objects with a form of parallelepiped in the world. People usually do not think about it, but architecture and various massive structures consist of several faces. Looks like parallelepiped can differently depend on the type.

Basic concepts and classification

The definition of parallelepiped, pyramids, cube and other polyhedra were known since ancient times. The main characteristics are simplicity and significance.

The derived V and S formulas are significant to solve various tasks with practical content and proof by theorems (according to drawings). Views of parallelepiped:

Parallelepiped tasks
  1. Straight. Four side faces have corners of 90 degrees.
  2. Rectangular. Each side of the figure is rectangular.
  3. Inclined.
  4. Dihedral, triangular. Consists of several faces at an angle of 90 degrees.
  5. Inclined, diagonal. Side faces are not perpendicular to the grounds.
  6. Rombohedron. The parties are equal diamonds.
  7. Cube Paralylepiped with equal (square) sides.

In the 6th grade in the geometry lesson, planimetry is studied (flat figures). Here is the scan of planes.

The two sides of the parallelepiped, who do not have a common rib are called opposite, and containing a single line - adjacent. From the point of view of planes, located in parallel, the three of their pairs intersect inside. These vertices connect the segment - diagonal. The length of the three edges of the correct polyhedron is called measurement . The main condition is the total peak.

When solving tasks, the concept of height is perpendicular, lowered from any vertex on the opposite direction. The face that the height falls is considered to be the grounds. Par Allepiped properties:

  • Any parties are parallelograms (with symmetry);
  • The parties located against each other will be parallel and equal.
Properties of parallelepipeda

Brick - an excellent example of a rectangular parallelepipeda (PP) . Also, its shape has nine-storey panel houses, bullfirers, wardrobes, containers for storing products and other household items.

The surface diagonals intersect and this central point is divided into several parts. They are equal to d2 = a2 + b2 + c2

The faces of the parallelepiped in front and rear are equivalent, as well as the upper and lower sides, but are not equal, because they are not opposite, but adjacent.

Formulas and analysis

For PP, it is true that its volume is equal to the magnitude of the triple product of the vectors of the three sides emanating from a single vertex. Formulas for PP:

All about parallelepiped
  1. V = a * b * c.
  2. S b = 2 * C * (A + B).
  3. S n = 2 * (A * B + B * C + A * C).

Decoding designations: V is the volume of the figure, S - surface area, A - length, B - width, C - height.

A special case of parallelepipeda, in which all sides are squares, is a cube. If any of the parties indicate the letter A, then formulas are used for the surface and volume: S = 6 * A * 2, V = 3 * A. In them v - the volume of the figure, A - the length of the face.

Parallelepipeda rules

The last variety of parallelepiped is a direct type. Its base will be parallelograms, and the base of PP is a rectangle. Formulas used in mathematics and geometry: SB = PO * H, SP = SB + 2SO, V = SO * H.

To find the answers, not enough to know only the properties of the geometric shape. Formulas may be useful for calculating S and V.

The PP diagonal is equal to the addition of the squares of its measurements: D2 = A2 + B2 + C2. This formula is obtained from the Pythagorean theorem.

ΔBad is rectangular, therefore Bd2 = AB2 + AD2 = B2 + C2 .

ΔBDD1 is rectangular, it means BD12 = BD2 + DD12. You need to substitute the value: d2 = a2 + b2 + c2.

Standard formula: V = SOSN * H. Decoding designations: V - the volume of parallelepiped, SOSN - the base area, H is height.

S is also the same as a parallelogram or rectangle. When solving tests and exam tasks it is easier to calculate the indicators of the prism, which is based on a straight angle. The formula for calculating the side of the parallelepiped SBOK = P * H can also be useful, where:

Tasks with parallelepiped
  • SBOK - Par Allepiped Square;
  • P - perimeter;
  • H is the height, perpendicular to the base.

The volume of the figure is equal to the magnitude of the mixed product of several vectors released from a single point.

Practical use

To calculate the volume, height and other characteristics of the figure you need to know theoretical foundations and formulas. The problem of tasks is included in the program of passing the exam and tickets upon admission to the university.

Proof Theorem

Theoretically s side surface of PP is equal to s b. p. = 2 (a + b) c. S full surface is equal to SP0. Surfaces PP = 2 (AB + AC + BC).

The volume of PP is equal to the product of three sidewalls overlooking a single vertex (three dimensions of PP): ABC.

Proof: Since PP side ribs perpendicular to the base, then they are its heights - H = aa1 = c. If a rectangle lies at the base, then SOSN = AB ⋅ ad = AB. Diagonal D PP can be found according to the formula D2 = A2 + B2 + C2, where A, B, C - measurements of PP.

If a rectangle is located at the base, then △ ABD rectangular, it means that the Pythagores Theorem BD2 = AB2 + AD2 = A2 + B2. If all side faces are perpendicular to the main line, then BB1 ⊥ (ABC) ⇒ BB1 ⊥ BD .

When △ BB1D is rectangular, then by the Pythagore Theorem B1d = BB12 + BD2.

Solving tasks

Parallelepiped photo

Task 1: PP: 3, 4, 12 cm are known, it is necessary to find the length of the main diagonal of the figure.

The search for an answer to the question begins with building a schematic image on which values ​​are meaning. The formula B1D2 = AB2 + AD2 + AA12 is used. After calculations, the expression B2 = 169, B = 13 is obtained.

Task 2: PP ribs emerging from a common point are equal to 3 and 4, total S - 94. You need to find the third edge coming out of the same vertex.

Ribs are indicated A1 and A2, and unknown - A3. The surface area is expressed s = 2 (A1A2 + A1A3 + A2A3).

Next, we obtain A3 (A1 + A2) = S / 2 - A1A2. Unknown Rib: A3 = S / 2 - A1A2 / A1 + A2 = 47-12 / 7 = 5.

Task 3: Two rectangular parallelepiped ribs coming out of a common point are 72 and 18, the diagonal is 78. It is necessary to determine the volume of the shape.

To solve, it is required to find a diagonal according to the formula for calculating the square root from the sum (A2 + B2 + C2), where A, B, C - the ribs of the shape. 78 - Root from the amount of 722 + 182 + C2. Decision:

Facts about parallelepiped
  • 78 = Root from the amount of 5508 + C2
  • 782 = 5508 + C2
  • C2 = 6084 - 5508.
  • C2 = 576.

Answer: The volume is 576.

Task 4: The edge of the inclined parallelepiped is 10 cm, the KLNM rectangle with measurements 5 and 7 cm is a cross section of the figure parallel to the edge. It is necessary to determine the side surface area of ​​the prism.

Kl and AD are not equal as a pair of ML and DC. Side S figures are equivalent to S section, multiplied by AA1, as the edge perpendicular to the cross section. Answer: 240 cm².

Task 5: abcda1b1c1d1 = 3, 4 cm, lateral edge - 12 cm. You need to determine the diagonal of PP.

Based on a rectangle with the sides of AB 3 cm and AD 4 cm. The side edge is 3 cm. BB1 is the height of PP and equals 12 cm. Diagonal B1d2 = AB2 + BB1 2 + = 9 + 16 + 144 = 169. B1d = 13 cm .

Task 6: The base of the PP is the square, one of the tops of its top base is equally removed from all vertices of the lower part. It is necessary to find the height of the shape if the base diagonal is 8 cm, and the side edge is 5 cm.

Basic concepts of parallelepipeda

One of the vertices of the base (F) is equivalent to removed from all vertices of the lower base of the parallelepiped. Together with the diagonal of the lower part (AC), it forms an equally chaired ΔAFC. AF = AC by condition. AF is an edge of the figure.

In an equilibried ΔAFC side the sides are the same: AF = FC = 5 cm, AC = 8 cm. The height ΔAFC will be the height of the parallelepiped.

The height of the triangle divides its base in half. By the Pythagore Theorem, it is equal to:

  • FK2 + (AC / 2) 2 = FC2;
  • FK2 + 16 = 25;
  • FK2 = 25-16 = 9;
  • FK = 3 cm.

The height of the figure is 3 cm.

The established theorems, evidence, as well as the derived formulas help calculate different values ​​for the figure.

In this publication, we will consider the definition, elements, types and basic properties of parallelepiped, incl. rectangular. The information provided is accompanied by visual drawings for better perception.

Definition of parallelepipeda

Parallelepiped - this is a geometric figure in space; Hexagon, whose faces are parallelograms. The figure has 12 ribs and 6 faces.

Parallelepiped

The parallelepiped is a variation of the prism with a parallelogram as a base. The main elements of the figures are the same as the prism.

Note: Formulas for calculating surface area (for a rectangular figure) and the volume of parallelepiped are presented in separate publications.

Views of parallelepiped

  1. Direct parallelepiped - The side faces of the shape are perpendicular to its bases and are rectangles. Direct parallelepiped
  2. Direct parallelepiped can be rectangular - The grounds are rectangles. Rectangular parallelepiped
  3. Inclined parallelepiped - Side faces are not perpendicular to the grounds. Inclined parallelepiped
  4. Cubic - All the edges of the shapes are equal squares. Cubic
  5. If all the faces of the parallelepiped are the same diamonds, it is called Rombohedron .

Properties of parallelepipeda

1. The opposite faces of the parallelepiped are mutually parallel and are equal to parallelograms.

2. All diagonals of the parallelepiped intersect at one point and are divided into it in half.

Diagonally parallelepipeda

3. Square diagonal (D) Rectangular parallelepipeda is equal to the sum of the squares of its three dimensions: length (a) , widths (b) and height (c) .

Diagonal of parallelepipedad2= A. 2+ B. 2+ C. 2

Note: To the parallelepiped, also applicable prism properties.

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