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Vesica Piscis — Definition, Formula & Examples

A vesica piscis is the almond-shaped region formed where two circles of equal radius overlap, with each circle's center lying on the other circle's circumference.

Given two circles of radius rr whose centers are separated by a distance equal to rr, the vesica piscis is their intersection — a lens-shaped region whose minor axis equals rr and whose major axis equals r3r\sqrt{3}.

Key Formula

A=r22(2π/3−3)×2=r2(2π3−3)A = \frac{r^2}{2}\left(2\pi/3 - \sqrt{3}\right) \times 2 = r^2\left(\frac{2\pi}{3} - \sqrt{3}\right)
Where:
  • AA = Area of the vesica piscis
  • rr = Radius of each circle (also the distance between centers)

How It Works

To construct a vesica piscis, draw a circle of radius rr. Then place a second circle of the same radius so that its center sits exactly on the first circle's circumference, and vice versa. The overlapping region is the vesica piscis. Its pointed ends lie along the line connecting the two centers, and its widest span runs perpendicular to that line. The figure has a height-to-width ratio of 3:1\sqrt{3} : 1, approximately 1.732:11.732 : 1.

Worked Example

Problem: Two circles each have radius 6 cm, and each circle passes through the other's center. Find the area of the vesica piscis they form.
Identify values: Both circles have radius r=6r = 6 cm, and the distance between centers equals r=6r = 6 cm.
Apply the area formula: Substitute r=6r = 6 into the vesica piscis area formula.
A=r2 ⁣(2π3−3)=36 ⁣(2π3−3)A = r^2\!\left(\frac{2\pi}{3} - \sqrt{3}\right) = 36\!\left(\frac{2\pi}{3} - \sqrt{3}\right)
Compute: Evaluate the expression numerically.
A=36(2.0944−1.7321)≈36(0.3623)≈13.04 cm2A = 36(2.0944 - 1.7321) \approx 36(0.3623) \approx 13.04 \text{ cm}^2
Answer: The vesica piscis has an area of approximately 13.0413.04 cm².

Why It Matters

The vesica piscis appears in Gothic architecture, stained-glass window design, and Venn diagrams. Understanding it strengthens your ability to compute areas of circular intersections, a skill used in probability, engineering tolerances, and computer graphics collision detection.

Common Mistakes

Mistake: Computing the area as a single circular segment instead of two.
Correction: The vesica piscis is the union of two equal circular segments, one from each circle. You must double the single-segment area or use the combined formula A=r2(2π/3−3)A = r^2(2\pi/3 - \sqrt{3}).