A pentagon egg is a planar egg formed by four tangent circular arcs constructed from a regular pentagon. Dumoutier (2021) describes the construction, which Johnny Ball also demonstrates (Numberphile 2022).
Label the vertices of the outer regular pentagon 
, 
, 
, 
, and 
clockwise from the top vertex.
Draw the five diagonals to form a pentagram.
Label the vertices of the inner regular
pentagon 
, 
, 
, 
, and 
clockwise from its upper left. Extend its diagonals 
and 
to meet sides 
and 
at 
and 
, respectively. Let 
be the intersection of 
and 
. Let 
and 
be the intersections of 
with 
and 
, respectively.
Draw the lower semicircle from 
to 
centered at 
. Draw the minor circular
arcs from 
to 
centered at 
, from 
to 
centered at 
, and from 
to 
centered at 
. At each join, the common point
and the two centers are collinear,
so the circular arcs are tangent.
If the outer regular pentagon has side length 
and 
is the golden ratio, the
semicircle, each side circular
arc, and the top circular arc have radii 
, 
, and 
, respectively. Their central
angles are 
, 
, and 
. The four chords
bound a trapezoid with parallel sides of lengths 
and 
and height 
. Adding its area
to those of the four circular segments gives
the enclosed planar area 
, with
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(OEIS A401885).
