In Euclidean geometry, a parallelogram is a simple (non self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean Parallel Postulate and neither condition can be proven without appealing to the Euclidean Parallel Postulate or one of its equivalent formulations. The three-dimensional counterpart of a parallelogram is a parallelepiped.
The etymology (in Greek παραλληλ-όγραμμον, a shape "of parallel lines") reflects the definition.
The perimeter of a parallelogram is 2(a + b) where a and b are the lengths of adjacent sides.
The sum of the distances from any interior point of a parallelogram to the sides is independent of the location of the point. (This is an extension of Viviani's theorem). The converse also holds: If the sum of the distances from a point in the interior of a quadrilateral to the sides is independent of the location of the point, then the quadrilateral is a parallelogram.
A parallelogram with base b and height h can be divided into a trapezoid and a right triangle, and rearranged into a rectangle, as shown in the figure to the left. This means that the area of a parallelogram is the same as that of a rectangle with the same base and height.
A parallelogram can be rearranged into a rectangle with the same area.
The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram
The area K of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles.
The area of the rectangle is
and the area of a single orange triangle is
Therefore, the area of the parallelogram is
Another area formula, for two sides B and C and angle θ, is
The area of a parallelogram with sides B and C (B ≠ C) and angle at the intersection of the diagonals is given by
When the parallelogram is specified from the lengths B and C of two adjacent sides together with the length D1 of (any) one diagonal, then the area can be found from Heron's formula. Specifically it is
where and the leading factor 2 comes from the fact that the number of congruent triangles that the chosen diagonal divides the parallelogram into is two.
The area on coordinate system
Let vectors and let denote the matrix with elements of a and b. Then the area of the parallelogram generated by a and b is equal to .
Let vectors and let Then the area of the parallelogram generated by a and b is equal to .
Let points . Then the area of the parallelogram with vertices at a, b and c is equivalent to the absolute value of the determinant of a matrix built using a, b and c as rows with the last column padded using ones as follows:
Proof that diagonals bisect each other
To prove that the diagonals of a parallelogram bisect each other, we will use congruenttriangles:
(alternate interior angles are equal in measure)
(alternate interior angles are equal in measure).
(since these are angles that a transversal makes with parallel linesAB and DC).
Also, side AB is equal in length to side DC, since opposite sides of a parallelogram are equal in length.
Therefore triangles ABE and CDE are congruent (ASA postulate, two corresponding angles and the included side).
Since the diagonals AC and BD divide each other into segments of equal length, the diagonals bisect each other.
Separately, since the diagonals AC and BD bisect each other at point E, point E is the midpoint of each diagonal.