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Parallelogram -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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Parallelogram

A parallelogram is a quadrilateral with opposite sides parallel (and therefore opposite angles equal). A quadrilateral with equal sides is called a rhombus, and a parallelogram whose angles are all right angles is called a rectangle. And, since a square is a degenerate case of a rectangle, both squares and rectangles are special types of parallelograms.

The polygon diagonals of a parallelogram bisect each other (Casey 1888, p. 2).

The angles of a parallelogram satisfy the identities

and

A+B=180 degrees.

(3)

A parallelogram of base b and height h has area

A=bh=absinA=absinB.

(4)

The height of a parallelogram is

h=asinA=asinB,

(5)

and the polygon diagonals p and q are

(Beyer 1987).

The sides a, b, c, d and diagonals p, q of a parallelogram satisfy

p^2+q^2=2(a^2+b^2)

(10)

(Casey 1888, p. 22).

The area of the parallelogram with sides formed by the vectors u=(u_x,u_y) and v=(v_x,v_y) is

where uxv is the two-dimensional cross product and detA is the determinant.

ParallelogramTheorem

As shown by Euclid, if lines parallel to the sides are drawn through any point on a diagonal of a parallelogram, then the parallelograms not containing segments of that diagonal are equal in area (and conversely), so in the above figure, A_1=A_2 (Johnson 1929).

ParallelogramSquares

The centers of four squares erected either internally or externally on the sides of a parallelograms are the vertices of a square (Yaglom 1962, pp. 96-97; Coxeter and Greitzer 1967, p. 84).

See also

Diamond, Lozenge, Medial Parallelogram, Parallelogram Illusion, Parallelogram Law, Quadrilateral, Rectangle, Rhombus, Square, Trapezoid, Varignon Parallelogram, Wittenbauer's Parallelogram Explore this topic in the MathWorld classroom

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References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 123, 1987.Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 84, 1967.Harris, J. W. and Stocker, H. "Parallelogram." §3.6.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 83, 1998.Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 3, 1948.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 61, 1929.Yaglom, I. M. Geometric Transformations I. New York: Random House, pp. 96-97, 1962.

Cite this as:

Weisstein, Eric W. "Parallelogram." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Parallelogram.html

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