Astroid Area Calculator

About this calculator

An astroid is the four-cusped curve traced by a point on a circle of radius a/4 rolling inside a fixed circle of radius a, defined parametrically as x = a·cos³t, y = a·sin³t (or implicitly as x^(2/3) + y^(2/3) = a^(2/3)). Its enclosed area is exactly (3/8)πa² and its total perimeter is exactly 6a — both remarkably clean closed forms despite the curve's non-circular, star-like shape.

This calculator takes the astroid's radius parameter a and returns its enclosed area and perimeter. Math students and educators use it to verify calculus derivations (the area and arc-length integrals are classic exercises), and engineers working with hypocycloid gear profiles and four-bar linkage mechanisms, where astroid-like envelope curves appear in cam and gear design, use it as a quick geometric reference.