CL=cut length if polygon side is 1
2 pieces 3 pieces 4 pieces
 T5 2008
 5 pieces
6 pieces
 T2
 T3
 T4

CL=0.6734+ CL=0.8660+ CL=1.3051+ CL=1.5803+ CL=1.7321+
 S4
 S2 1

 S5
 S6 2008
CL=1.0000 CL=1.6233+ CL=1.9755+ CL=2.5021+ CL=2.9399+
 P3
 P4
 P5
 P6
CL=1.4310+ CL=2.1587+ CL=2.8340+ CL=3.3383+ CL=3.7420+
 H2
 H3
 H4
 H5
 H6
CL=1.7321+ CL=2.5981+ CL=3.4641+ CL=4.3301+ CL=4.9137+
 Hp4 Hp6 Hp3 Hp2 Hp5

CL=1.8351+ CL=3.1828+ CL=4.1800+ CL=5.1985+ CL=5.7635+
 Principles for constructing equal area dissections with minimal cut length   ° straight lines or circle arcs   ° vertices with only 3 edges   ° equal angle vertices, 120°   ° joining border with right angle   ° Laplace rule: sum of signed curvatures at vertex is zero
 I got very valuable hints from Warren D. Smith, 13.8.2007. The findings can nicely be verified with soap bubbles with well defined confinement and monolayers. See one picture for S6 from J.Reber.
Warren D. Smith
Zoomable pictures in pdf
 T5b
 T5c
 T5d arcs
CL=1.6166+ ? CL=1.6145+ CL=1.3817+ CL=1.6037+
 P2b
 P4a
 P3a
 S4a
CL=1.3422+ CL=1.5388+ CL=2.8485+ CL=2.2057+ CL=2.4192+
 T4c Hall
 P4c
 P4b
 P4e
 S6a
CL=1.3173+ ? CL=2.9699+ CL=2.8587+ CL=2.9484+
 P5a
 P5b
 P5c
 P2a
 S6b
CL=3.4410+ CL=3.5337+ CL=3.3892+ CL=1.4553+ CL=3.0000
 H5a
 H5b
 H6b
 S6c
CL=2.9942+ CL=4.3541+ CL=4.3808+ ? ?
 H6a
 H6d
 H7a
 H4a
 Hp4a
CL=5.1962+ CL=5.1962+ CL=5.1962+ CL=3.7321+ CL=4.2201+
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