Diagonal

Diagonal ( from the ancient Greek διά dia. "Through" and γωνία gonia " corner, angle" ) is a term used in geometry. Generally, the term certain routes that connect vertices of faces or bodies together; for the precise definition see below.

  • 2.1 Number of diagonals
  • 2.2 lengths of the diagonals

Diagonals in the plane geometry

In the planar geometry is referred to as the diagonal links of non-adjacent vertices in a polygon (polygon ), which, therefore, must have at least four corners.

Number of diagonals

The number d of the diagonals of an n- corner, that is in a polygon having the number of corners n is.

For each of the n vertices is with ( n - 3) connected by a diagonal corners (not with themselves and not with the two neighboring corners). By the denominator ( divisor) 2 in the formula is taken into consideration that with this approach over all vertices of each diagonal would be generated twice in a complete revolution.

The result for the vertex set of 3 to 25:

For a convex polygon all diagonals lie entirely within the polygon, with a concave polygon at least one diagonal completely outside.

Lengths of the diagonals

Generally

The diagonal length d from one corner to the next corner is calculated s0 from the length of two intermediate pages and s1 and the intervening angle φ1 according to the law of cosines:

Are at a diagonal for a partial circumference between the ends of the diagonal of the sides and the interior angles of the intermediate vertices are known, then the diagonal length can be calculated from it.

Denoting by a diagonal starting end, the sides si and each lying in front with interior angle? I then:

  • For the diagonal across three pages:
  • For the diagonal across four pages applies:
  • Generally for a diagonal across n pages:

Special cases

For some special cases, there are simple formulas for the diagonal length:

  • The diagonal length of a parallelogram with sides a and b and the inner angles are α:

And

  • Therefore applies to the diagonal length of a rectangle ( φ = 90 °) with the side lengths a and b according to the Pythagorean theorem:

Regular Polygons

  • The diagonal length of a square with side length a can be calculated according to
  • For the diagonal length of a regular pentagon with side length a
  • For the diagonals on two or three pages in a regular hexagon with side length a

Diagonals in the room geometry

In the solid geometry is meant by the diagonal of a polyhedron such a route that connects two corners of the body to each other, but not with an edge even with a diagonal of a side surface coincides ( room or body diagonal).

Number of diagonals

To find the number of diagonals of a polyhedron, subtract the number of corners the value 1 from multiplying the remainder by the number of vertices and halved the product. The number thus obtained is first subtracts the number of all edges and the diagonals of all the number of side faces.

Denoting the number of corners with, the number of faces with the number of edges and the number of vertices of the surface with a No. i we obtain

For all parallelepipeds, for example, Cuboid, results with

Lengths of the diagonals

  • For the special case of the cube results from it.

Application in the arts

In the visual arts diagonals are used as a compositional element for the formal design of works. In particular, the diagonal method divides a rectangular format in two squares and then uses the diagonals of the image array.

Perception

Diagonals in the broader sense of art and ( photo ) graphic design are perceived differently: those who will rise in the course right up received as a positive and dynamic and therefore referred to as a positive diagonal. At the opposite pole is the negative diagonal of the coming from the upper left running in the picture to the bottom right - symbol for brake, stop negative feelings. Feature of good design it can be to have a diagonal or diagonal structure, the image statement supportive direction, possibly faced with a small version of its counterpart, a counter- diagonal. Explained the effect is the psychology of perception of the writing and the reading direction to the right, to the right upwardly extending orientation of fast hand-written letters, italic printing type, according to the forearm movement to the right above the rising lines of handwriting and further illustrations from science to market price if on the current to the right timeline to an applied upward value to develop positively.

Planimetrics Italic type
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