Vesica Piscis

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The Vesica Piscis is the convex intersection of two congruent circles that pass through each other's center. Here we take the common radius as 2 and the centers on the y-axis at (0,±1). Since each circle from which S is constructed is symmetric with respect to the y-axis and each is reflected across the x-axis from the other, S is bisymmetric. In order for S to be traversed in a counterclockwise direction around the origin starting from its right-hand corner, its parametric equations are:
The abscissa extrema of S are at and the ordinate extrema at the two circle centers (0,±1), so the width x height of the bounding rectangle of S is .


The perimeter of the Vesica Piscis is 8π/3 and its area is

Boundary Circles & Circumellipse

The squared distance function x2(t) + y2(t) of S ranges from a minimum of 1 at (0,±1) to a maximum of 3 at , so the inradius is 1 and the circumradius is . Since these extrema occur on the coordinate axes, a candidate for the circumellipse is one enclosed by the annulus between the two boundary circles, with a = and b = 1. For verification, we have


In Lemma B, we have for the upper arch
x(t)y(t) = 2cos(2t/3 + π/6)[2sin(2t/3 + π/6) - 1]
This expression is maximized in the first quadrant at the complicated value
The circumellipse dimensions are
For verification, we have

Summary Table

IncircleR = 16.2831853.141593
Vesica Piscis8.3775804.913479
CircumcircleR = 10.8827969.424778
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The Vesica Piscis (red) is a member of a group of similarly-shaped figures described on these pages, including (inside to outside) the mouth curve and the cycloid:

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