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Squircle — Definition, Formula & Examples

A squircle is a shape that looks like a blend between a square and a circle, with smoothly rounded corners but flatter sides than a true circle.

A squircle is the specific superellipse defined by the equation x4+y4=r4x^4 + y^4 = r^4, where rr is the radius. It is a special case of the Lamé curve ∣x∣n+∣y∣n=rn|x|^n + |y|^n = r^n with exponent n=4n = 4.

Key Formula

x4+y4=r4x^4 + y^4 = r^4
Where:
  • xx = Horizontal coordinate
  • yy = Vertical coordinate
  • rr = Radius — the distance from center to the midpoint of each side

How It Works

The squircle equation x4+y4=r4x^4 + y^4 = r^4 works similarly to the circle equation x2+y2=r2x^2 + y^2 = r^2, but with fourth powers instead of second powers. Raising coordinates to a higher power pushes the curve outward toward a square shape while keeping the corners smooth. As the exponent nn increases in ∣x∣n+∣y∣n=rn|x|^n + |y|^n = r^n, the shape approaches a perfect square. At n=2n = 2 you get a circle, at n=4n = 4 a squircle, and as n→∞n \to \infty a square.

Worked Example

Problem: Verify that the point (1,1)(1, 1) lies inside, on, or outside the squircle x4+y4=2x^4 + y^4 = 2.
Identify r: The equation is x4+y4=2x^4 + y^4 = 2, so r4=2r^4 = 2 and r=21/4≈1.189r = 2^{1/4} \approx 1.189.
Substitute the point: Plug in x=1x = 1 and y=1y = 1.
14+14=1+1=21^4 + 1^4 = 1 + 1 = 2
Compare to r⁴: Since 1+1=2=r41 + 1 = 2 = r^4, the point satisfies the equation exactly.
Answer: The point (1,1)(1, 1) lies exactly on the squircle x4+y4=2x^4 + y^4 = 2.

Why It Matters

Squircles appear in product design and UI engineering — Apple's app icons use a shape closely related to a squircle for aesthetically pleasing rounded corners. In mathematics, studying superellipses like the squircle connects algebra, coordinate geometry, and the concept of LpL^p norms used in linear algebra and analysis.

Common Mistakes

Mistake: Assuming a squircle is just a rounded square (a square with circular-arc corners).
Correction: A rounded square is made of straight edges joined by quarter-circle arcs. A squircle is a single smooth algebraic curve with no straight segments and continuously changing curvature.