ConverterHub

Astroid Area Calculator

Enter the astroid's radius parameter a to find its enclosed area and perimeter.

How it is calculated

The astroid is defined by x2/3 + y2/3 = a2/3, where a is the distance from the centre to each of the four cusps (the pointed tips). Integrating the curve gives two closed forms, which the tool uses: area = (3/8)·π·a² and curve length = 6a. The length is that of the whole outline, all four arcs together. The value of a must be above zero, and its unit carries through: lengths come out in the same unit as a, areas in that unit squared. Both results are shown with four decimals.

Worked example

An astroid has its cusps 10 units from the centre, so a = 10. Area: (3/8) × π × 10² = 0.375 × π × 100 ≈ 117.81, which the tool shows as 117.8097 square units. Curve length: 6 × 10 = 60, shown as 60.0000 units. For comparison, a circle of radius 10 has an area of 314.16, so the astroid fills 37.5% of that circle.

Try this example →

Common mistakes

  • Entering the full width of the shape. The parameter a is the distance from the centre to a cusp, so the whole astroid spans 2a across, and the width must be halved first.
  • Using the circle formula π·a². The astroid is the star-shaped curve inside that circle and holds only 3/8 of its area.
  • Expecting the length to involve π. The length of the curve is exactly 6a, with no π.
  • Mixing the area unit and the length unit. The area is in square units, the length in plain units.
  • Using the formula for a different hypocycloid. Curves with a different number of cusps have other formulas; this one is only for the four-cusped astroid.

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.

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