which of the shape is equalable shape
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Answer:
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Step-by-step explanation:
Equal content and equal shape, figures of
Two figures in having equal area and corresponding to two polygons and that can be decomposed into polygons so that the parts making up are congruent, respectively, to the parts making up .
For , , equality of content means equality of volume; equal shape for polyhedra is defined similarly to . These ideas have been generalized to non-Euclidean geometry also.
Area (of a polygon) is a function satisfying the following axioms:
) for any polygon ;
) if is a pairwise-disjoint (up to boundary points) union of polygons , then
) if and are congruent, then ;
) the area of a square with sides of unit length is 1.
Using these axioms the area of a rectangle can be determined.
Contents
1 Theorem.
1.1 References
1.2 Comments
1.3 References
Theorem.
If two polygons have equal shape, then they have equal area.
Based on this theorem is the method of subdivision, known as long ago as Euclid: To calculate the area of a polygon one attempts to divide it into a finite number of parts from which it is possible to make up a figure of known area. For example, a parallelogram is of equal shape with a rectangle of the same base and height (see Fig. a); a triangle is of equal shape with a parallelogram of the same base and half the height (see Fig. b).
Figure: e035900a
Figure: e035900b
Thus, the complete theory of areas of polygons can be constructed on the basis of the theorem on the area of a rectangle.
There is another approach to the calculation of areas that is based on the axioms ) and ) — the complementation method. Two polygons are called equal by complementation if corresponding congruent parts can be adjoined to them so that congruent polygons are obtained. For example, parallelograms and rectangles with the same bases and heights are equal by complementation (see Fig. c), and hence are of equal area.
Figure: e035900c
In the Euclidean plane, two polygons are of equal area if and only if they are of equal shape (and also if and only if they are equal by complementation). A similar theorem is valid in the Lobachevskii plane and in the elliptic plane. On the other hand, in non-Archimedean geometry only "being of equal area" and "being equal by complementation" are equivalent; "being of equal shape" is not equivalent to them.
The theory of volumes in is based on analogues of the axioms )–) for area. However, for the calculation of the volume of a tetrahedron, from Euclid on, a limit transition (the "devil's staircasedevil's staircase" ) has been used, and, in modern textbooks, an integral, the definition of which is also related to a limit, is used. The foundation for the use of a "superfluous" (in comparison to planimetry) limit, the proof that it is impossible to calculate the volume of an arbitrary tetrahedron by the methods of subdivision and complementation, formed Hilbert's third problem. In 1900 M. Dehn solved the third problem by proving that a regular tetrahedron and a cube of equal area are not of equal shape. For two polyhedra and of equal area to be of equal shape it is necessary and sufficient that the Dehn invariant (a function of the lengths of the edges and the sizes of the corresponding dihedral angles, see [2]) satisfies .
There are multi-dimensional generalizations of Dehn's invariant, in which necessary conditions for being of equal shape have been formulated, and it has been proved that for a regular -dimensional simplex is not of equal shape to a cube of equal area. In a necessary condition for being of equal shape is also sufficient.
Let be a group of motions of the plane. Two polygons and are called -congruent if there is a motion such that . Two polygons and are said to be of -equal shape if they can be subdivided into parts in such a way that the parts making up are -congruent to the corresponding parts making up . For polyhedra, being of -equal shape is defined similarly.
Let be the group of motions consisting of all parallel translations and central symmetries. The notions of being of equal shape and being of -equal shape are equivalent in . In particular, polygons of equal area can be subdivided into parts so that their respective parts are not only congruent but have corresponding parallel side.