Calculations at an astroid, a hypocycloid with four cusps. An astroid is the path of a circle with the radius a/4 (diameter = a/2) inside another circle with the radius a. Enter one value and choose the number of decimal places. Then click Calculate.
Formulas:
pi:
Radius, length and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
The astroid is a two-dimensional, concave shape composed of four identical curved lines that are not circular arcs. A corresponding shape formed from four circular arcs is referred to here as a circular-arc quadrangle. The length of a single arc of the astroid is one-quarter of its total perimeter, 3/2 a. So both such an arc and the area enclosed by the arcs are relatively simple to calculate, which is a characteristic that is by no means a given for curved shapes not based on circular arcs.
The curve of an astroid can be described by the equations x=a*cos³(t) and y=a*sin³(t), where the angle t ranges from 0 to 2π. Here, as above, the parameter a is the radius of the fixed circle within which the generating circle rolls.
The astroid has four axes of symmetry. Two of these pass through pairs of opposite vertices, and two pass through the midpoints of pairs of opposite circular arcs. These two axes intersect at the center of the astroid. With respect to this point of intersection, the astroid possesses point symmetry as well as rotational symmetry for rotations of 90 degrees and its multiples.
The astroid appears in various fields of mathematics and physics, particularly wherever rolling motions or hulls play a role. It can be understood as the hull of a family of line segments of fixed length whose endpoints lie on two mutually perpendicular lines. In physics, it appears, among other contexts, in the description of caustics, which are curves of light formed through reflection or refraction. In technical applications, such as in specialized cam mechanisms or gear profiles, the astroid can serve as an idealized trajectory.