Calculate Diagonals in Area and Space
Calculator for the diagonals in a rectangular area and in a cuboid-shaped space. A diagonal is a line between two vertices that are not on the same edge. Its length in two and in three dimensions can be calculated here. For a diagonal in an area, enter two values, for one in a space, enter three values.
The formula is d = √ a² + b² + c²
Area diagonal (red) and space diagonal (blue). The space is spanned by a cuboid, whose faces are rectangles.
The length of a diagonal of a rectangle is calculated using the Pythagorean theorem. Here, the diagonal is the hypotenuse, and the two opposite sides of the rectangle are the legs (or catheti). A diagonally bisected rectangle is a right triangle, to which the Pythagorean theorem applies. The formula d = √ a² + b² + c² is the extended Pythagorean theorem for three dimensions, or the space Pythagorean theorem. This length is also called the Euclidean distance.
Diagonals can, but don't necessarily have to, be located between opposite vertices. Only in a quadrilateral do they always do so. In a cuboid, there are two different types of diagonals. Those of the quadrilateral, i.e., the diagonals of the face, and the space diagonals. Space diagonals in cuboids are always located between opposite vertices, while face diagonals are not, related to space. Examples of two-dimensional diagonals that are not located between opposite vertices are the diagonals in a pentagon, which has no opposite vertices, and the short diagonals in a hexagon.
The word diagonal comes from ancient Greek and means through corners or angles. In Euclidean geometry, diagonals are straight lines. In non-Euclidean geometries, these lines appear curved from the outside. The Pythagorean theorem and the space Pythagorean theorem do not apply in these geometries.
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