Antiparallelogram Calculator

Name: Antiparallelogram - Geometry Calculator
Author: Jürgen Kummer

Calculations at an antiparallelogram, a crossed quadrilateral of two congruent triangles. An antiparallelogram has two pairs of opposite sides of equal length. The longer sides b intersect each other and divide into the sections p and q.
Enter the length of the short side a and two of the three values b, p and q. Choose the number of decimal places and click Calculate. Angles are calculated and displayed in degrees, here you can convert angle units.

Antiparallelogram shape for p≥q (if p<q, just switch both values):

Formulas:

b > a

b = p + q

α = \arccos (\frac{p^{2} + q^{2} - a^{2}}{2 p q})

β = \arccos (\frac{a^{2} + q^{2} - p^{2}}{2 a q})

γ = \arccos (\frac{a^{2} + p^{2} - q^{2}}{2 a p})

δ = 180 ° - α

s = \sqrt{2 p^{2} - \cos (180 ° - α) \cdot 2 p^{2}}

t = \sqrt{2 q^{2} - \cos (180 ° - α) \cdot 2 q^{2}}

h = \sqrt{a^{2} - {(\frac{s - t}{2})}^{2}}

p = 2 \cdot (a + b)

A = 2 \cdot \sqrt{\frac{p}{4} \cdot (\frac{p}{4} - a) \cdot (\frac{p}{4} - p) \cdot (\frac{p}{4} - q)}

Lengths, chords, height and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

The antiparallelogram is nonconvex, it is axially symmetric to the bisecting line of the outer angles δ.

Like the parallelogram, the antiparallelogram has two pairs of sides of equal length. The sides of one pair are opposite each other, those of the other pair intersect. The angles also form two equal pairs. The two pairs of sides are antiparallel to each other. Antiparallelism is a relatively complicated geometric property for pairs of straight lines and angles, where equal angles are not opposite but adjacent.
A parallelogram can be formed from an antiparallelogram by reflecting one of the two triangles along the perpendicular to the axis of symmetry through their intersection and then connecting the adjacent points of the four corners. Conversely, but more complicated, an antiparallelogram can be formed from a parallelogram by drawing the diagonals, removing two opposite sides and then mirroring one of the triangles and rotating it accordingly. The parallelogram and antiparallelogram then have one pair of equal and one pair of different lengths.

Constructions based on an antiparallelogram are used in mechanics and kinematics. One example is four-bar linkages on a crank arm. These allow flexible deformation while maintaining the length ratios of the construction.

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