Calculations at an arrowhead quadrilateral. Such a quadrilateral is also called a concave deltoid or dart.
Enter one side a or b and two of the three angles α, β and γ, choose the number of decimal places and click Calculate. Please enter angles in degrees, here you can convert angle units. The angle β must be larger than 180°.
Formulas:
α + β + 2 * γ = 360°
a / sin( β / 2 ) = b / sin( α / 2 )
m = a * sin( γ ) / sin( β / 2 )
n = b * sin ( β / 2 - 90° )
l = m + n
h = 2 * √ b² - n²
p = 2 * ( a + b )
A = l * h / 2 - n * h / 2
Lengths, height and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
The arrowhead quadrilateral is axially symmetrical to the line on which the inner length m and the missing length n lie. It consists of two mirror-image triangles with sides a, m, b and a, b, m respectively, where the angle between m and b (or m and a) must be greater than 90 degrees for the quadrilateral to be concave. With exactly 90 degrees you would get an isosceles triangle, with less you would get a convex kite. The arrowhead quadrilateral has at least two acute angles, these are the two equal, opposite angles γ. The angle at the arrowhead α can be acute or obtuse. The angle at the concave corner β is superobtuse, i.e. it has more than 180 degrees, since it is made up of two angles that are both greater than 90 degrees.
Sometimes the arrowhead quadrilateral is counted among the kites or deltoids, sometimes not. There is no precise rule here and it depends on the exact definition of kites, which can vary depending on the source. It tends to belong to this category, but if this is not desired, it should be explicitly excluded, for example by a restriction to convex deltoids. The common between the convex and the concave shape is the equal length of adjacent pairs of sides and a pair of equal, opposite angles.