You will need

- Knowledge of the lengths of all sides of a parallelogram.

Instruction

1

Method 1. Given a rectangular parallelepiped with sides a, b, c and diagonal d. According to one of the properties of a parallelogram, the square of the diagonal is equal to the sum of the squares of its three sides. It follows that the length of the diagonal can be calculated by extracting the square of this amount (Fig.1).

2

Method 2. Suppose a rectangular parallelepiped is a cube. Cube - is a cuboid whose each face is represented by a square. Consequently, all its sides are equal. Then the formula to calculate the length of its diagonal is expressed as:

d = a*√3

d = a*√3

# 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

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

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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

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

V=a^3

# Advice 4: How to calculate the diagonal of the rectangle

A closed geometric figure formed by two pairs lying opposite each other parallel lines of equal length is called a parallelogram. And a parallelogram, all angles of which equal 90°, it is called a rectangle. In this figure it is possible to hold two lines of equal length connecting opposite vertices is a diagonal. The length of these diagonals is computed in several ways.

Instruction

1

If you know the lengths of two adjacent sides

**of the rectangle**(A and b), then the length of the diagonal (C) determine very simple. Assume that**the diagonal**is opposite the right angle in the triangle formed by it and these two parties. This allows you to apply in the calculation of the Pythagorean theorem to calculate the diagonal length by finding the square root of the sum of squared lengths of the known sides: C=v(A?+In?).2

If you know the length of only one side

**of the rectangle**(A) and the angle (?), which with it forms a**diagonal**, to calculate the length of this diagonal (S) will have to use one of the direct trigonometric functions - cosine. Divide the length of the known sides into the cosine of the known angle - this will be the required length of the diagonal: C=A/cos(?).3

If the rectangle specified by the coordinates of its vertices, the problem of computing the length of its diagonal is reduced to finding the distance between two points in this coordinate system. Apply the Pythagorean theorem to the triangle formed by the projection of the diagonal on each of the coordinate axes. For example, a rectangle in two-dimensional coordinates formed by the vertices A(X?;Y?), B(X?;Y?), C(X?;Y?) and D(X?;Y?). Then you need to calculate the distance between points a and C. the Length of the projection of this segment on the X-axis will be equal to the difference between the coordinates of |X?-X?|, and the projection on the Y - axis |Y?-Y?|. The angle between axes is 90°, which implies that these two projections are the legs, and the diagonal length (hypotenuse) is equal to the square root of the sum of squares of their lengths: AC=v((X?-X?)?+(Y?-Y?)?).

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For finding the diagonal

**of a rectangle**in three-dimensional coordinate system proceed as in the previous step, only adding to the formula the length of the projection on the third coordinate axis: AC=v((X?-X?)?+(Y?-Y?)?+(Z?-Z?)?).# Advice 5: How to calculate the diagonal of the rectangle

In memory left many mathematical joke: Pythagorean pants on all sides are equal. Use it to calculate

**the diagonal****of a rectangle**.You will need

- A sheet of paper, ruler, pencil, calculator with function of calculate roots.

Instruction

1

A rectangle is a quadrilateral, all the angles which are straight. The diagonal

**of the rectangle**is the line segment connecting two opposite vertices.2

*On a sheet of paper with a ruler and pencil draw an arbitrary rectangle ABCD. Better to do it on notebook sheet in a cage – it will be easier to draw right angles. Connect the cut vertices*

**of the rectangle**A and C. the resulting segment AC is a

**diagonal**th

**rectangle**ABCD.

3

Please note,

**the diagonal**AC has divided the rectangle ABCD into the triangles ABC and АСD. The resulting triangles ABC and АСD straight triangles, because the angles ABC and АDС is 90 degrees (by definition**of a rectangle**). Remember the Pythagorean theorem – the square of the hypotenuse is equal to the sum of the squares of the other two sides.4

The hypotenuse is the side of the triangle opposite the right angle. Sides – the sides of the triangle adjacent to the right angle. With respect to the triangles ABC and АСD: AB and BC, AD and DC sides of the AC common hypotenuse for both triangles (the desired

**diagonal**). Therefore, as in a square = square AB + square BC or AC square = AD square + square DC. Input the side lengths**of the rectangle**in the above formula and calculate the length of the hypotenuse (diagonal**of rectangle**).5

For example, the sides

**of the rectangle**ABCD is equal to the following values: AB = 5 cm and BC = 7cm. the square of the diagonal AC of this**rectangle**is calculated by the Pythagorean theorem: AC square = AB square + a square sun = 52+72 = 25 + 49 = 74 sq cm using the calculator calculate the value of square root of 74. You should have 8.6 cm (rounded value). Keep in mind that one of the properties**of a rectangle**, its diagonals are equal. So the length of the second diagonal BD**of the rectangle**ABCD is equal to the length of the diagonal AC. For the above example this value is 8.6 cm